Some boundary element methods for heat conduction problems - Jultika

SOME BOUNDARY ELEMENT METHODS FOR HEAT CONDUCTION PROBLEMS

M AR T T I HAMINA Mathematics Division

OULU 2000

MARTTI HAMINA

SOME BOUNDARY ELEMENT METHODS FOR HEAT CONDUCTION PROBLEMS

Academic Dissertation to be presented with the assent of the Faculty of Science, University of Oulu, for public discussion in Raahensali (Auditorium L 10), Linnanmaa, on June 21st, 2000, at 12 noon.

O U L U N Y L I O P I S TO , O U L U 2 0 0 0

Copyright © 2000 Oulu University Library, 2000

Manuscript received 28 March 2000 Accepted 12 April 2000

Communicated by Professor Arvet Pedas Professor Rolf Stenberg

ISBN 951-42-5614-X

ALSO AVAILABLE IN PRINTED FORMAT ISBN 951-42-5613-1 ISSN 0355-3191

(URL: http://herkules.oulu.fi/issn03553191/)

OULU UNIVERSITY LIBRARY OULU 2000

Hamina Martti, Some boundary element methods for heat conduction problems Mathematics Division, University of Oulu, P.O. Box 4500, FIN 90014 Oulu, Finland 2000 Oulu, Finland (Manuscript received 28 March 2000)

Abstract This thesis summarizes certain boundary element methods applied to some initial and boundary value problems. Our model problem is the two-dimensional homogeneous heat conduction problem with vanishing initial data. We use the heat potential representation of the solution. The given boundary conditions, as well as the choice of the representation formula, yield various boundary integral equations. For the sake of simplicity, we use the direct boundary integral approach, where the unknown boundary density appearing in the boundary integral equation is a quantity of physical meaning. We consider two different sets of boundary conditions, the Dirichlet problem, where the boundary temperature is given and the Neumann problem, where the heat flux across the boundary is given. Even a nonlinear Neumann condition satisfying certain monotonicity and growth conditions is possible. The approach yields a nonlinear boundary integral equation of the second kind. In the stationary case, the model problem reduces to a potential problem with a nonlinear Neumann condition. We use the spaces of smoothest splines as trial functions. The nonlinearity is approximated by using the L2 -orthogonal projection. The resulting collocation scheme retains the optimal L2 -convergence. Numerical experiments are in agreement with this result. This approach generalizes to the time dependent case. The trial functions are tensor products of piecewise linear and piecewise constant splines. The proposed projection method uses interpolation with respect to the space variable and the orthogonal projection with respect to the time variable. Compared to the Galerkin method, this approach simplifies the realization of the discrete matrix equations. In addition, the rate of the convergence is of optimal order. On the other hand, the Dirichlet problem, where the boundary temperature is given, leads to a single layer heat operator equation of the first kind. In the first approach, we use tensor products of piecewise linear splines as trial functions with collocation at the nodal points. Stability and suboptimal L2 -convergence of the method were proved in the case of a circular domain. Numerical experiments indicate the expected quadratic L2 -convergence. Later, a Petrov-Galerkin approach was proposed, where the trial functions were tensor products of piecewise linear and piecewise constant splines. The resulting approximative scheme is stable and convergent. The analysis has been carried out in the cases of the single layer heat operator and the hypersingular heat operator. The rate of the convergence with respect to the L2 -norm is also here of suboptimal order.

Keywords: collocation, boundary integrals, heat conduction

Acknowledgements The basis of this research was laid during the years 1989-1991 when I was working as a researcher in the project: ”Analysis of the Boundary Element Method”. The project was funded by the Academy of Finland. Professor Jukka Saranen was the head of our team. I wish to express my gratitude to the Academy of Finland for financial support. This research was carried out under the supervision of Professor Jukka Saranen. His encouragement has been of great importance during the evolution of my work. I am deeply indebted to my instructor for his guidance and advice. I also wish to express my thanks to Dr Keijo Ruotsalainen for his collaboration. I am also very grateful to Professor Rolf Stenberg and Professor Arvet Pedas for refereeing the manuscript. The linguistic form of the English language was revised by Mr Gordon Roberts, to whom I express my thanks for his contribution. The Mathematics Division in the Faculty of Technology at the University of Oulu has been a stimulating working environment. For this I am grateful to the staff of the Division. In particular, I wish to express my thanks to Professor Juhani Nieminen and Professor Seppo Seikkala for their encouragement during this work. Finally, I wish to express my warmest thanks to my parents for their constant support.

Oulu, March 2000

Martti Hamina

List of the original articles This thesis is a summary of the work published in the following five articles: I Hamina M & Saranen J (1994) On the spline collocation method for the single-layer heat operator equation. Math Comp 62: 41-64. II Hamina M (1997) A collocation type projection method for the single layer heat operator equation. Preprint, University of Oulu, December 1997. III Hamina M (2000) An approximation method for the hypersingular heat operator equation. J Comput Appl Math 115: 229-243. IV Hamina M, Ruotsalainen K & Saranen J (1992) The numerical approximation of the solution of a nonlinear boundary integral equation with the collocation method. J Integral Equations Appl 4: 95-115. V Hamina M (1997) On the numerical solution of a non-linear heat conduction problem. Integral Methods in Science and Engineering, Vol 2, (Constanda C, Saranen J, Seikkala S eds), Addison Wesley Longman Inc: 93-98. These articles are referred to in the text by their Roman numerals.

Contents Abstract Acknowledgements List of the original articles Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Recent history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Boundary integral formulations . . . . . . . . . . . . . . . . . . . . 1.3 Reduction to an integral equation . . . . . . . . . . . . . . . . . . . 1.4 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Mapping properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Spline spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Inverse properties . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Approximation properties . . . . . . . . . . . . . . . . . . . 1.7 Some Petrov-Galerkin approximations . . . . . . . . . . . . . . . . 1.7.1 Reformulation of the collocation problem . . . . . . . . . . 1.7.2 Reformulation of the Petrov-Galerkin method . . . . . . . . 1.8 Description of the nonlinearity . . . . . . . . . . . . . . . . . . . . 2 Abstract theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Summary of the original articles . . . . . . . . . . . . . . . . . . . . . . 3.1 Article I: On the spline collocation method for the single layer heat operator equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Article II: A collocation type projection method for the single layer heat operator equation . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Article III: An approximation method for the hypersingular heat operator equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Article IV: The numerical approximation of the solution of a nonlinear boundary integral equation with the collocation method . . 3.5 Article V: On the numerical solution of a non-linear heat conduction problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 14 17 19 21 23 24 25 27 29 30 33 36 36 38 42 42 45 47 50 52 54

1 Introduction Initial and boundary value problems are seldom analytically solvable. Therefore numerical methods are of great importance. The Finite Element Method has been among the most popular computational techniques. However, in some instances the Boundary Element Method has evolved as a powerful alternative for finite elements. Compared to the finite element method, the most important feature of the boundary element method is that it only requires discretization of the boundary rather than that of the whole volume. Thus, the boundary element method is advantageous when the domain extends to infinity or the shape of the boundary is complex. The most severe restriction of the boundary element approach is that it is applicable only to such problems where the fundamental solution is available. The boundary and initial value problem of the heat equation is the most usual model problem of the so-called parabolic boundary and initial value problems. This work is devoted to the boundary element solution of the homogeneous heat equation. An exellent classical treatment of partial differential equations of the parabolic type is given by Friedman [22]. Pogorzelski’s book [45] serves as a classical introduction to integral equation methods. Some textbooks concerning the modern treatment of integral equation methods and boundary element methods are Atkinson [8], and Chen & Zhou [13].

1.1 Recent history The mathematical theory of the boundary element method has developed vastly during the last twenty years. The most familiar discretization methods are the spline Galerkin and spline collocation methods. Other well known approaches are the Nystr¨ om method, trigonometric collocation and quadrature methods. During 1976-1983 a rather complete theory of the Galerkin boundary element method on smooth boundaries was developed. The theory was based on the concept of strongly elliptic pseudodifferential operators. Thus, the analysis covered very large classes of boundary element methods. The practical implementation, as well as

12 the effect of the numerical integration, has been discussed in references [26], [27] and [35], for example. In the presence of singularities (domains with corners, mixed boundary conditions) theoretical progress has been based on the works of Costabel, Stephan, Chandler and several other authors. In the Galerkin approach, the boundary integral equation is satisfied in the average sense. From the practical point of view, this means double integration over the boundary of the domain. Another approach, easier to implement, is the collocation method where the boundary integral equation is fulfilled on certain, so-called, collocation points. However, the theoretical analysis of the collocation method is much more difficult. There are two main lines of research. One is based on the equivalence between the collocation method and a certain Galerkin method. The other relies on direct Fourier analysis. The equivalent Galerkin method is available only when the trial functions are splines of an odd degree. The analysis allows quasi-uniform meshes and yields asymptotic error estimates with respect to some nonstandard Sobolev norms, which depend on the order of the operator and on the degree of the trial functions. The method was developed by Arnold & Wendland [5] for the one-dimensional case and extended by Arnold & Saranen [4] for biperiodic problems. These ideas generalize also to the case of certain parabolic boundary integral equations, as was shown by Hamina & Saranen in [24], [25] and paper I. The direct Fourier analysis is applicable if the meshes are uniform. This approach has been applied to the one-dimensional spline collocation in papers [6], [56], [54]. We emphasize that the convergence of the spline collocation in higher dimensions is not generally known for quasi-uniform meshes, even in the case of elliptic problems. There are results in some special cases where the equation is given on a torus [4], [15], [47], square or cube [17], [46]. Recently Costabel & Saranen [18] have obtained optimal order error estimates for convolutional parabolic boundary integral equations by using Fourier techniques. For time dependent problems, the papers by Arnold & Noon [3] and Noon [37] were of crucial importance. They discovered that the single layer heat operator is coercive in certain anisotropic Sobolev spaces. Their analysis applied to three dimensional heat conduction problems. Corresponding results in two space dimensions were given by Hsiao & Saranen [28] and Costabel [14]. The coercivity property was at once a rigorous basis for the analysis of Galerkin boundary element methods. The fundamental mapping properties of the associated boundary integral operators as well as the solvability theory was developed by Costabel [14] and Hsiao & Saranen [29]. The first theoretical result concerning the spline collocation approximation of the single layer heat operator equation was in a conference paper by Hamina & Saranen, [24]. The given results cover the case where the spatial domain is a disk. Later, Hamina & Saranen in paper I analysed a method where the trial functions were tensor products of piecewise linear splines defined on a quasi-uniform space-time mesh. Stability and suboptimal convergence of the method as well as numerical results were reported. The extension of the approach to more general boundary curves remained to be done. In their conference article, Häm¨ al¨ ainen & Saranen [32] reported a compact perturbation technique which generalizes the

13 analysis to more general boundary curves, provided that the parametrization is with respect to the arc length parameter. A complete discussion of this topic is presented by Hämäläinen [31]. In paper II Hamina suggested a Petrov-Galerkin approach, where interpolation was used with respect to the space variable, and orthogonal projection was used with respect to the time variable. He applied the method to the single layer heat operator equation as well as to the hypersingular heat operator equation. In both cases, quasi-uniform meshes are allowed. For the single layer heat operator equation (paper II), the trial functions were tensor products of piecewise linear (space) and piecewise constant splines. The hypersingular heat operator decreases smoothness. Therefore, numerical approximation of the solution of the hypersingular heat operator equation (paper III) requires a smoother boundary density. The lowest order trial functions are tensor products of piecewise cubic (space) and piecewise linear splines. In articles II and III, stability and suboptimal convergence of the resulting Petrov-Galerkin method was proved. In the case of the hypersingular heat operator equation, a stability condition between the mesh parameters appears in contrast to the case of the single layer heat operator equation which is unconditionally stable. It is remarkable here that, in the one-dimensional case, optimal order error estimates with respect to the L2 -norm are available, in contrast to the multidimensional and time-dependent cases where the proof of optimal order error estimates is not known. Another line of research started when Ruotsalainen & Wendland [53] converted a nonlinear Neumann boundary value problem for the Laplacian to a nonlinear boundary integral equation. They proved that the resulting nonlinear integral equation is uniquely solvable, and the spline Galerkin boundary element method is stable and convergent. Their approach was based on the theory of monotone operators. Later, the idea was extended to the collocation method by Ruotsalainen & Saranen [52], and to Lp -theory by Eggermont & Saranen [20]. In their conference paper Hamina, Ruotsalainen & Saranen [23] proposed an approximate collocation scheme such that the nonlinearity is replaced by its L2 -orthogonal projection. 3 1 An asymptotic error estimate of order O(h 2 ) with respect to the H 2 -norm was presented for the solution of the resulting approximate collocation equation. A numerical example was also given. The proofs, as well as several extensions, optimal order L2 -convergence, the effect of the numerical integration - were published in paper IV. The proof of the L2 -convergence is based on the results by Saranen [55]. Ruotsalainen [48] considered a nonlinear mixed boundary value problem for the Laplacian in the plane. He converted the problem into a system of boundary integral equations and proved quasi-optimality estimates for the Galerkin boundary element solutions for this nonlinear operator equation. In article [49] Ruotsalainen adapted the analysis to cases where the nonlinearity satisfies a polynomial growth condition. He considered the indirect formulation which transforms the boundary value problem for the Laplacian into a nonlinear boundary integral equation. The results on stability and optimal order error estimates for the Galerkin method in Lp spaces are given. The corresponding convergence results for the collocation

14 method are presented in Ruotsalainen [50]. Analogously to the elliptic case, the Neumann problem for the homogeneous heat equation with vanishing initial data can be reduced to a nonlinear boundary integral equation. Therefore, it was natural to extend the linear theory to certain nonlinear problems by applying the theory of monotone operators to boundary integral operators acting on anisotropic Sobolev spaces. Hsiao & Saranen [30] obtained stability and optimal order convergence for the Galerkin approximation. Ruotsalainen & Saranen [52] presented convergence results for the Galerkin approximation when the nonlinearity had a polynomial growth condition. In paper V Hamina proposed a method where interpolation is used with respect to the space variable and orthogonal projection is used with respect to the time variable. The trial functions are tensor products of piecewise linear (space) and piecewise constant splines. Hamina obtained stability and optimal order convergence for the resulting Petrov-Galerkin approximation. In contrast to the above mentioned Petrov-Galerkin method, stability and convergence of the collocation method is still an open problem.

1.2 Boundary integral formulations Considering the heat conduction problem, we first introduce a boundary integral approach for solution of the homogeneous heat equation with the given nonlinear Neumann type boundary condition and vanishing initial data. Let Ω ⊂ R2 be a bounded domain with the smooth boundary Γ = ∂Ω. With 0 < T < ∞, we have the heat conduction problem, (see [30], [52] and paper V)  in QT = Ω × (0, T ),  ∂t Φ − ∆Φ = 0, ∂n Φ(x, t) = −F (x, t, Φ(x, t)) + gN Γ (x, t), on ΣT = Γ × (0, T ), (1.1)  Φ(x, 0) = 0, x ∈ Ω. In the stationary case, the model problem reduces to the following potential equation with a given nonlinear Neumann type boundary condition (see [53], [51] and paper IV) ∆Φ = 0, in Ω, (1.2) ∂n Φ(x) = −F (x, Φ(x)) + gN Γ (x), on Γ. Another time dependent model problem is the homogeneous heat equation with the given Dirichlet type boundary condition and vanishing initial data (see [14], [29], and paper I)  in QT = Ω × (0, T ),  ∂t Φ − ∆Φ = 0, Φ(x) = gDΓ (x, t), on ΣT = Γ × (0, T ), (1.3)  Φ(x, 0) = 0, x ∈ Ω. The classical single layer and double layer heat potentials V and W are defined by the expressions

15

(Vσ)(x, t) =

Zt Z 0

(Wµ)(x, t) =

(1.4)

µ(y, τ )∂ny E(x − y, t − τ )dΓy dτ , ((x, t) ∈ QT ∪ QcT ),

(1.5)

Γ

Zt Z 0

σ(y, τ )E(x − y, t − τ )dΓy dτ , ((x, t) ∈ QT ∪ QcT ),

Γ

¯ ∂n is the exterior normal derivative and where QcT = Ωc × (0, T ), Ωc = R2 \Ω, y ( 2 1 − |x| 4t , t > 0, 4πt e E(x, t) = (1.6) 0, t ≤ 0, denotes the fundamental solution of the two-dimensional heat equation. For sufficiently smooth boundary densities σ and µ, the heat potential Φ(x, t) = (Vσ)(x, t) − (Wµ)(x, t),

((x, t) ∈ QT ),

(1.7)

combined with the boundary behaviour of the single layer and double layer potential and their normal derivatives ∂n V, ∂n W yields Φ|ΣT ∂n Φ|ΣT

= =

SΓ σ + ( 12 I − DΓ )µ ( 12 I + DΓ0 )σ + HΓ µ.

(1.8) (1.9)

This is, in fact, a compact way to write the well known boundary behaviour Vσ|ΣT Wµ|ΣT

= SΓ σ, = (DΓ − 12 I)µ,

(interior limit).

(1.10) (1.11)

Here, the boundary integral operators SΓ , DΓ , DΓ0 and HΓ with corresponding representations are the single layer heat operator (SΓ σ)(x, t) =

Zt Z 0

σ(y, τ )E(x − y, t − τ )dΓy dτ ,

(x ∈ Γ)

(1.12)

Γ

the double layer heat operator (DΓ µ)(x, t) =

Zt Z 0

µ(y, τ )∂ny E(x − y, t − τ )dΓy dτ ,

(x ∈ Γ)

(1.13)

(x ∈ Γ)

(1.14)

Γ

the spatial adjoint of the double layer heat operator (DΓ0 σ)(x, t)

=

Zt Z 0

Γ

σ(y, τ )∂nx E(x − y, t − τ )dΓy dτ ,

16 and the hypersingular heat operator (HΓ µ)(x, t)

= −∂nx

Zt Z 0

=

−

Zt Z 0

µ(y, τ )∂ny E(x − y, t − τ )dΓy dτ ,

(1.15)

Γ

µ(y, τ )∂nx ∂ny E(x − y, t − τ )dΓy dτ ,

(x ∈ Γ).

Γ

The direct boundary integral method is, in the case of the Dirichlet problem (1.3), based on the heat potential (1.16) Φ = Vσ − WgDΓ combined with the boundary relation (1.8). This approach yields the single layer heat operator equation of the first kind (see papers I and II) SΓ σ = ( 12 I + DΓ )gDΓ .

(1.17)

For the linear Neumann problem (1.1) with F = 0, the heat potential Φ = VgN Γ − Wµ

(1.18)

together with the boundary relation (1.9) yields the hypersingular heat operator equation of the first kind (see [29] and paper III) HΓ µ = ( 12 I − DΓ0 )gN Γ .

(1.19)

In the following, the Nemitsky operator is defined by F(µ)(x, t) = F (x, t, µ(x, t)). The properties of the nonlinearity are given later after fixing the spaces of functions. In the direct boundary integral method, the heat potential Φ in (1.1) is given by means of the representation Φ = V(gN Γ − F(µ)) − Wµ,

(1.20)

which by (1.10) and (1.11) yields the nonlinear boundary integral equation of the second kind, (see [30] and paper V) ( 12 I + DΓ )µ + SΓ F(µ) = SΓ gN Γ .

(1.21)

We observe that the linear case F = 0 reduces to the linear boundary integral equation of the second kind ( 12 I + DΓ )µ = SΓ gN Γ ,

(1.22)

which has a well established theory [14], [16], [29], [40]. In certain anisotropic Sobolev spaces, equations (1.17), (1.19), (1.21) and (1.22) are uniquely solvable and the solutions admit the interpretations Φ Σ = µ, (1.23) T ∂n Φ Σ = σ. (1.24) T

17 On the other hand, the indirect single layer potential approach Φ = V(˜ σ ),

(1.25)

yields for the boundary density the nonlinear boundary integral equation ( 12 I + DΓ0 )˜ σ + F(SΓ σ ˜ ) = gN Γ .

(1.26)

This approach has been applied to the case of the boundary value problem of the Laplacian with a nonlinear boundary condition [48], [49], [50] as well as to the case of the boundary and initial value problem of the heat equation [52]. In these articles, unique solvability of the resulting nonlinear operator equation has been established, as well as stability and convergence of the chosen numerical method in appropriate spaces of functions.

1.3 Reduction to an integral equation For linear problems, we perform the analysis in spaces of functions which are periodic with respect to the space variable. In the case of nonlinear problems, this reduction is not used. We assume that the boundary curve has the smooth, regular, one-periodic parametric representation x : R → Γ such that the Jacobian |x0 (θ)| is strictly positive. Let R2T = R × (0, T ). Then, the associated heat operators can be written as (Su)(θ, t)

=

Zt Z1

=

Zt Z1

=

Zt Z1

0

(Du)(θ, t)

0

(D0 u)(θ, t)

0

(Hu)(θ, t)

=

−

u(ϕ, τ )E(x(θ) − x(ϕ), t − τ )|x0 (ϕ)| dϕ dτ ,

0

u(ϕ, τ )

nϕ · (x(θ) − x(ϕ)) E(x(θ) − x(ϕ), t − τ )|x0 (ϕ)| dϕ dτ , 2(t − τ )

u(ϕ, τ )

nθ · (x(θ) − x(ϕ)) E(x(θ) − x(ϕ), t − τ )|x0 (ϕ)| dϕ dτ , 2(t − τ )

0

0

Zt Z1 0

u(ϕ, τ )∂n(θ) ∂n(ϕ) E(x(θ) − x(ϕ), t − τ )|x0 (ϕ)| dϕ dτ .

0

We emphasize that the kernel function corresponding to the hypersingular heat operator has strong singularity, and appropriate regularization methods are needed in actual numerical computations. Using the notations gD (θ, t) = gDΓ (x(θ), t), gN (θ, t) = gN Γ (x(θ), t), we obtain (SΓ σ)(x(θ), t) = (Su)(θ, t), with u(θ, t) = σ(x(θ), t) and (HΓ µ)(x(θ), t) = (Hu)(θ, t) with u(θ, t) = µ(x(θ), t). Thus, equations (1.17) and (1.19) on the

18 boundary transform to ordinary integral equations (Su)(θ, t)

=

(Hu)(θ, t)

=

1 2 gD (θ, t) + (DgD )(θ, t), 1 0 2 gN (θ, t) − (D gN )(θ, t),

(1.27) (1.28)

over the domain (θ, t) ∈ R × [0, T ]. We recall that sufficiently smooth functions, which are one-periodic with respect to variable θ, admit the Fourier representation Z 1 X u(θ, t) = u ˆ(n, η)ein2πθ+iηt dη, 2π n∈Z R

where the Fourier coefficient is defined by u ˆ(n, η) =

R1 R

u(θ, t)e−in2πθ−iηt dt dθ.

0 R

In terms of the Heaviside step function H, the heat kernel corresponding to the circular boundary of radius ρ, defined by 2 2 π(θ−ϕ) exp − ρ sin(t−τ ) Eρ (θ − ϕ, t − τ ) = H(t − τ ) , 4π(t − τ ) is a function of the difference of the coordinates. Therefore, the associated boundary integral operators Sρ , Dρ , Dρ0 and Hρ are of a convolutional type PR 1 s(n, η)ˆ u(n, η)ein2πθ+iηt dη, (1.29) (Sρ u)(θ, t) = 2π n∈Z R PR 1 (Dρ u)(θ, t) = 2π d(n, η)ˆ u(n, η)ein2πθ+iηt dη, (1.30) n∈Z R PR 0 1 (Dρ0 u)(θ, t) = 2π d (n, η)ˆ u(n, η)ein2πθ+iηt dη, (1.31) n∈Z R PR 1 (Hρ u)(θ, t) = 2π h(n, η)ˆ u(n, η)ein2πθ+iηt dη. (1.32) n∈Z R

As an example, we give here the explicit representation formula of the single layer heat operator 2 2 π(θ−ϕ) Zt Z1 exp − ρ sin(t−τ ) (Sρ u)(θ, t) = u(ϕ, τ ) ρ dϕ dτ 4π(t − τ ) 0

0

=

ρ 2π

+∞ R R1

=

ρ 2π

−∞ 0

PR

√ u ˆ(ϕ, η)K0 (2ρ iη sin π|θ − ϕ|) dϕ dη √ √ u(n, η)ein2πθ+iηt dη. In (ρ iη) Kn (ρ iη)ˆ

n∈Z R

Here In and Kn are the modified Bessel functions, [1]. See also [39]. For the hypersingular heat operator, representation (1.32) follows from the fact that the kernel 2 2 π(θ−ϕ) exp − ρ sint−τ ˜ − ϕ, t − τ ) = h(θ −∂n(θ) ∂n(ϕ) Eρ (θ − ϕ, t − τ ) = −∂n(θ) ∂n(ϕ) 4π(t − τ ) is a function of the difference of the coordinates. Similar considerations are true for Dρ and Dρ0 .

19

1.4 Spaces Let H s (Γ), s ∈ R (see [2]) be the usual Sobolev space equipped with the norm 1 R kuks = (u , u)s2 . In particular, L2 (Γ) = H 0 (Γ) and (u , v)0,Γ = Γ u(y)v(y)dΓy . For s > 0 the norm is given by m X

kuk2s =

kDk uk20 +

k=0

|Dm u(x) − Dm u(y)|2 dΓx dΓy , |x − y|1+2δ

Z Z Γ Γ

where s = m + δ, 0 < δ < 1 and the derivative is taken with respect to the arc length. The negative order Sobolev spaces are defined by duality with respect to the L2 - inner product. Theoretical analysis of the heat equation leads to the concept of anisotropic Sobolev spaces. Here, we concentrate on the anisotropic spaces on the space-time boundary. These are defined as intersection spaces ([33] p. 87, [34] p. 8). H r,s (ΣT ) = H 0 ((0, T ); H r (Γ)) ∩ H s ((0, T ); H 0 (Γ)),

(r, s ≥ 0).

In particular, we describe the vanishing initial condition in terms of the subspace r,s H00 (ΣT ) = {u | u = U |ΣT : U ∈ H r,s (ΣT ), U (·, t) = 0, t < 0},

(0 < T < ∞).

−r,−s The negative order spaces H00 (ΣT ) for 0 < r < 1, 0 < s < 12 are defined by −r,−s r,s 0 duality H00 (ΣT ) = (H00 (ΣT )) . Having introduced the parametric representation for the boundary curve, it is enough to consider functions which are one-periodic with respect to the spatial variable. First, for any r ∈ R, let H r be the Sobolev space of one-periodic functions on R. The anisotropic spaces are defined by

H r,s := H 0 (R; H r ) ∩ H s (R; H 0 ),

(r, s ≥ 0).

In terms of the Fourier transform with respect to the time variable Z∞

(Ft u)(θ, η) =

u(θ, t)e−iηt dt,

−∞

the norm of the space H r,s is given by kuk2H r,s

=

Z∞

(k(Ft u)(·, η)k2H r + |η|2s k(Ft u)(·, η)k2H 0 ) dη.

(1.33)

−∞

The space H r,s (R2T ), r, s ≥ 0, 0 < T ≤ ∞ is the space of restrictions to R2T of functions belonging to H r,s equipped with the infimum norm H r,s (R2T ) kukH r,s (R2T )

=

{ u | u = U |R2T : U ∈ H r,s },

= inf{ kU kH r,s | u = U |R2T }.

(1.34) (1.35)

20 Again, we have the Hilbert space H r,s (R2T ) = H 0 ((0, T ); H r ) ∩ H s ((0, T ); H 0 ),

(0 < T ≤ ∞)

endowed with the norm kuk2H r,s (R2 ) T

=

ZT

ku(·, t)k2H r dt + kuk2H s ((0,T );H 0 ) ,

(0 < T < ∞).

0

We also use equivalent norms defined by kuk2H r,s (R2 ) T

ZT

=

ku(·, t)k2H r

dt +

0

ZT ZT 0

0

dt +

Z1

ku(·, t) − u(·, τ )k2H 0 dt dτ , |t − τ |1+2s

(0 < s < 1)

and kuk2H r,s (R2 ) T

=

ZT

ku(·, t)k2H r

0

ku(θ, ·)k2H s (0,T ) dθ,

(0 ≤ s ≤ 1).

(1.36)

0

Moreover, we introduce the subspace r,s H00 (R2T ) = {u | u = U |R2T : U ∈ H r,s , U (·, t) = 0, t < 0},

(0 < T < ∞).

r,s Thus, for example H00 (R2∞ ) is the space of those functions in H r,s (R2∞ ) for which the zero extension with respect to the time variable remains in H r,s . Finally, we −r,−s need the negative order space H00 (R2T ) for 0 < r < 1, 0 < s < 12 , which is −r,−s r,s defined by duality H00 (R2T ) = (H00 (R2T ))0 . Next, we consider some sets of continuous (respectively smooth) functions, which are one-periodic with respect to the first argument. Let C1 (R2 ) (resp. C1∞ (R2 )) be the space of continuous (resp. infinitely smooth) functions. The space ∞ C10 (R2 ) (resp. C10 (R2 )) consists of continuous (resp. infinitely smooth) functions u, which are defined on R2 , such that the support supp u(θ, ·) ⊂ K, where K ⊂ R is a compact set. The spaces of restrictions C1 (R2T ) (resp. C1∞ (R2T )) consist of continuous (resp. infinitely smooth) functions such that u = U |R2T . The space C10 (R2T ) ∞ (resp. C10 (R2T )) consists of continuous (resp. infinitely smooth) functions u, which are defined on R2T , such that the support supp u(θ, ·) ⊂ K, where K ⊂ R is a compact set. In order to describe the initial condition, we introduce the subspace ∞ C00 (R2T ) = {u | u = U |R2T : U ∈ C1∞ (R2 ), U (·, t) = 0, t < 0},

(0 < T ≤ ∞).

r,s ∞ This subspace enjoys the property that the embedding C00 (R2T ) ⊂ H00 (R2T ) is dense for all r, s ∈ R, if s ≥ 0. A comprehensive proof of the following Sobolev embedding theorem is given in [31] or [32]. For the convenience of the reader the proof is repeated here.

Theorem 1 In the anisotropic spaces the embeddings r

H r, 2 (R2 )

⊂

C(R2 ),

r

2

H r, 2 (R2T ) ⊂ C(RT ),

are continuous provided that r > 23 .

r, r

2

H002 (R2T ) ⊂ C00 (RT )

21 r

r

Proof. We present the proof for H r, 2 = H r, 2 (R2 ). Note that the anisotropic norm (1.33) is given by XZ 1 ˆ (n, η)|2 dη, (U ∈ H r, r2 ). (1 + |n| + |η| 2 )2r |U kU k2H r, r2 = n∈Z R

r

∞ Since C10 (R2 ) is dense in H r, 2 , r ≥ 0, it is enough to present the proof for U ∈ ∞ 2 C10 (R ). The Cauchy-Schwarz inequality applied to the Fourier representation Z 1 X ˆ (n, η)ein2πθ+iηt dη U (θ, t) = U 2π n∈Z R

gives |U (θ, t)| ≤ =

≤

Z 1 X ˆ (n, η)| dη |U 2π n∈Z R Z 1 1 1 X ˆ (n, η)| dη (1 + |n| + |η| 2 )−r (1 + |n| + |η| 2 )r |U 2π n∈Z R v v Z uX Z 1 u 1 1 u uX −2r ˆ (n, η)|2 dη. 2 (1 + |n| + |η| ) t (1 + |n| + |η| 2 )2r |U t 2π n∈Z R

n∈Z R

The result follows, since the function I(r) =

PR

1

(1 + |n| + |η| 2 )−2r is bounded,

n∈Z R

if r > 32 . The other embedding results follow by applying the infimum characterization of the norm for the subspaces of restrictions. The anisotropic Sobolev spaces enjoy a similar compactness property as ordinary Sobolev spaces. Theorem 2 Assume that 0 ≤ r1 ≤ r and 0 ≤ s1 ≤ s. Then the embedding r,s r1 ,s1 H00 (R2T ) ⊂ H00 (R2T ) is compact for any finite T > 0. The proof is given in article [29], Theorem 3.1 p. 93.

1.5 Mapping properties The convergence analysis and error estimates are based on the mapping properties of the heat operators in the anisotropic Sobolev spaces. In this work, we just quote the results. The proofs are lengthy and technical, and can be found in the given articles. Throughout this work c, c0 , ci etc. are generic constants, which may change in size, but are always independent of the mesh parameters. In the following, the duality pairing extends the L2 (R2T )-inner product.

22 r+1, r+1 2

r, r

Theorem 3 The single layer heat operator S : H002 (R2T ) → H00 isomorphism for all r ≥ − 12 . Furthermore, it is coercive such that (Sw , w) ≥ c kwk2− 1 ,− 1 2

4

− 1 ,− 14

for all w ∈ H002

(R2T ) is an

(R2T ).

(1.37)

For the proof see [3], [37], [14], [28]. The mapping properties for the double layer heat operator and its spatial adjoint are given in Theorem 4 The mapping properties of the operators D and D0 are: r

(i) D, D0 : H r, 2 (R2T ) → H r+1,

r+1 2

(R2T ) are bounded for all r ≥ − 12 .

r

r

(ii) D, D0 : H r, 2 (R2T ) → H r, 2 (R2T ) are compact for all r ≥ − 12 . r

r

(iii) For any a 6= 0, r ≥ − 12 the operators aI+D, aI+D0 : H r, 2 (R2T ) → H r, 2 (R2T ) are isomorphisms. This result is proved in paper [29], Theorem 4.2. See also [14], Corollary 3.14. r−1, r−1 2

r, r

Theorem 5 The hypersingular heat operator H : H002 (R2T ) → H00 an isomorphism for r ≥ 12 . Furthermore, it is coercive such that 1 1 2,4 Hw , w ≥ c kwk21 , 1 for all w ∈ H00 (R2T ).

(R2T ) is (1.38)

2 4

The proof is given in [29], Theorem 4.4 or [14], Corollary 3.13. The following coercivity estimate is used for the analysis of the nonlinear boundary integral equation. The proof can be found in [30], Theorem 4.1. 1 1

,

− 1 ,− 1

2 4 Theorem 6 The operator SΓ−1 ( 12 I + DΓ ) : H00 (ΣT ) → H002 4 (ΣT ) is an isomorphism. Further, it is strongly coercive, i.e. there exists a positive constant c such that 1 1 2,4 SΓ−1 ( 12 I + DΓ )w , w Σ ≥ c kwk21 , 1 for all w ∈ H00 (ΣT ). (1.39) T

2 4

The kernel functions of the heat operators are, in the case of circular boundaries, functions of the difference of the coordinates. Therefore, the heat operators formally commute withR the partial differentation operator. Also, the mean value 1 functional (J w)(t) := 0 w(θ, t) dθ commutes with the heat operators. r, r

Theorem 7 Let w ∈ H002 (R2T ). For r ≥ s − 12 , ∂θs (Sρ w)(θ, t) = (Sρ (∂ϕs w))(θ, t), s = 1, 2, . . . For r ≥ s +

1 2,

∂θs (Hρ w)(θ, t)

=

(Hρ (∂ϕs w))(θ, t),

s = 1, 2, . . .

(1.40) (1.41)

Moreover, J (Sρ w)(t) = Sρ (J w)(t) and J (Hρ w)(t) = Hρ (J w)(t).(1.42) For the moment, we adopt some notations and results from [31] and [32]. Let ρ S◦ be the single layer heat operator for the circle with radius 2π . We define ∂θ

:=

∂θ + J,

Kw Sw

:= :=

(S − S◦ )w, S◦ w + Kw.

Kw = ∂ θ K∂ −1 θ ,

23 Theorem 8 Assume that the one-periodic parametric representation of the boundary curve is chosen such that |x0 (θ)| = ρ = constant > 0. Let 0 < T < ∞. Then the operator S admits the representation S = ∂ θ S∂ −1 θ , and we have the mapping properties − 1 ,− 14

S : H002

r, r2

1 1

,

2 4 (R2T ) → H00 (R2T ) is an isomorphism,

r+2, r+2 2

K : H00 (R2T ) → H00

(R2T ) continuous, if r ≥ − 12 .

The proof of these results can be found in [31], Section 3. It is based on explicit estimation of the symbol of the perturbation operator K. The continuity properties of the operator K combined with the compactness result given in Theorem 2, show that the operator K can be interpreted as a compact perturbation of the principal part S◦ . Thus, a standard theory of compact perturbations is available. We emphasize that this technique generalizes the analysis of our Petrov-Galerkin methods to general, smooth boundary curves with the restriction that the parametric representation is with respect to the arc length parameter. However, these aspects are not explicitly shown here.

1.6 Spline spaces Next, we discuss the space of approximations. Let 0 = θ0 < θ1 < . . . < θN = 1 be a one-periodic mesh, with h = max{θn+1 − θn }. The mesh is called quasi-uniform if max{θn+1 − θn } ≤ C, min{θn+1 − θn } where C ≥ 1 is a constant. We denote by S dhθ ([0, 1]) the space of one-periodic, smoothest splines of degree dθ . Note that S dhθ ([0, 1]) is a N -dimensional linear space with basis functions {ψndθ }N n=1 . Analogously, we define the quasi-uniform grid 0 = t0 < t1 < . . . < tM = T , k = max{tm+1 − tm }. With respect to the t time variable, we consider the lowest order spaces S dk,[0,T ] . In practice, these are the spaces of piecewise constant splines generated by the basis functions (m = 1, . . . , M ) 1, if tm−1 < t < tm , 0 φm (t) = 0, otherwise, or the spaces of piecewise linear continuous splines generated by the basis functions (m = 1, . . . , M )  t−tm  1 + tm −tm−1 , if tm−1 < t < tm , t−tm 1 1 − tm+1 −tm , if tm ≤ t < tm+1 , φm (t) =  0, otherwise. We draw the attention of the reader to the fact that the elements of the space ϕ ∈ S 1k,[0,T ] satisfy the initial condition ϕ(0) = 0. The trial functions on the

24 space-time boundary are tensor products M X N X dθ ,dt t v(θ, t) = Sh,k (R2T ) := S dhθ ([0, 1]) × S dk,[0,T = v αm,n ψndθ (θ)φdmt (t) , ] m=1 n=1

dt M where {ψndθ }N n=1 and {φm }m=1 are the one-dimensional basis functions. We notice that the inclusion r, r

dθ ,dt (R2T ) ⊂ H002 (R2T ), Sh,k

(r < min{dθ + 12 , 2dt + 1}),

is generally valid for the tensor product spline spaces. Our methods use pointwise values. Therefore, it is essential that the images Su and Hu are continuous. In the light of the mapping properties combined with the Sobolev embedding theorem, we obtain appropriate ranges for Sobolev indices 1 2 5 2

< r < min{dθ + 12 , 2dt + 1} < r < min{dθ +

1 2 , 2dt

+ 1}

for the single layer heat operator, for the hypersingular heat operator.

1.6.1 Inverse properties We now recall some basic inverse and approximation results. For quasi-uniform meshes {θn } and {tm }, the one-dimensional inverse estimates kψkr kφkr kφkr

≤ c h−(r−s) kψks , ≤ c k −(r−s) kφks , ≤ c k −(r−s) kφks ,

(ψ ∈ Shdθ , (φ ∈ S 0k,[0,T ] , (φ ∈ S 1k,[0,T ] ,

are available [21]. In the anisotropic spaces, the valid kvkr, r2 ≤ c max h−r , k −r/2 kvk0,0 , kvk0,0 ≤ c max h−r , k −r/2 kvk−r,− r2 , kvkr, r2 ≤ c max h−r , k −r/2 kvk0,0 , kvk0,0 ≤ c max h−r , k −r/2 kvk−r,− r2 , kvkr, r2 ≤ c max h−r , k −r/2 kvk0,0 , kvk0,0 ≤ c max h−r , k −r/2 kvk−r,− r2 .

s ≤ r < dθ + 21 ), s ≤ r < 12 ), s ≤ r ≤ 1),

following inverse estimates are 0,0 (v ∈ Sh,k , 0 ≤ r < 21 ), (1.43) 0,0 (v ∈ Sh,k , 0 ≤ r ≤ 12 ), (1.44) 1,0 (v ∈ Sh,k , 0 ≤ r < 1), (1.45) 1,0 (v ∈ Sh,k , 0 ≤ r < 1), (1.46) 1,1 (v ∈ Sh,k , 0 ≤ r < 1), (1.47) 1,1 (v ∈ Sh,k , 0 ≤ r < 1).

(1.48)

Note that estimates (1.43) and (1.44), as well as the other two pairs, can be combined to a single estimate. For the sake of simplicity, the use of two estimates is preferred. We recall here the proof given in paper I, Lemma 5.2 p. 56-57.

25 1,1 Proof. Let v ∈ Sh,k and 0 ≤ r < 1. The representation formula (1.36) of the anisotropic norm, together with one-dimensional inverse estimates yields (1.47)

kvk2r, r

2

RT

≤c

kv(·, t)k2H r dt + c

0

R1

kv(θ, ·)k2

r

H 2 (0,T )

0

dθ

R1 c h−2r kv(·, t)k2H 0 dt + c k −r kv(θ, ·)k2H 0 (0,T ) dθ 0 0 r 2 ≤ c h−2r + k −r kvk20,0 ≤ c max( h−r , k − 2 ) kvk20,0 .

≤

RT

−r,− r2

We use the duality H00 inequality implies kvk20,0

0 r, r (R2T ) = H002 (R2T ) for 0 ≤ r < 1. Then, the Schwarz r

≤ kvkr, r2 kvk−r,− r2 ≤ c max( h−r , k − 2 )kvk−r,− r2 kvk0,0 ,

which yields (1.48). The inverse estimates (1.45), (1.46) for 0 ≤ r < 1 and (1.43), (1.44) for 0 ≤ r < 21 follow analogously. To see (1.44) for r = 12 we use 1

1

kvk0,0 ≤ c max( h− 4 , k − 8 )kvk− 14 ,− 18 ,

0,0 (v ∈ Sh,k ),

(1.49)

which further gives kvk2− 1 ,− 1 4

8

≤ kvk0,0 kvk− 12 ,− 14 1

1

≤ c max( h− 4 , k − 8 )kvk− 14 ,− 18 kvk− 12 ,− 14 ,

and therefore

1

1

kvk− 14 ,− 18 ≤ c max( h− 4 , k − 8 )kvk− 12 ,− 14 . Using this with (1.49) we obtain (1.44) with r = 12 .

1.6.2 Approximation properties The associated one-dimensional orthogonal projection operator P hdθ : H 0 → Shdθ is defined by (P hdθ u , v) = (u , v) for all v ∈ Shdθ . The following approximation result is true for periodic splines, [21], ku − P hdθ ukH r

≤ c hs−r kukH s ,

(u ∈ H s ,

0 ≤ r ≤ s ≤ dθ + 1, r < dθ + 12 ).

With respect to the time variable, the orthogonal projection P kdt : H 0 (0, T ) → t S dk,[0,T ] is defined analogously. Now, we restrict ourselves to the approximation properties ku − P k0 ukH r (0,T ) ku − P k1 ukH r (0,T )

≤ c k s−r kukH s (0,T ) , ≤ c k s−r kukH s (0,T ) ,

(u ∈ H s (0, T ), 0 ≤ r ≤ s ≤ 2, r < 12 ) (u(0) = 0, u ∈ H s (0, T ), 0 ≤ r ≤ 1).

26 In the piecewise linear case, u satisfies the initial condition u(0) = 0 and s = 1 or s = 2. For r = 0 or r = 1, the result is a standard error bound for the piecewise linear case. The interpolation inequality yields the given estimate for 0 < r < 1. Note that in the piecewise constant case, no initial condition is required. In the following, we use the same notation for two-dimensional and one-dimensional interpolation and projection operators e.g. P hdθ = P hdθ ⊗ I. dθ ,dt dθ ,dt The two-dimensional L2 -projection P h, : L2 (R2T ) → Sh,k (R2T ) is defined k dθ ,dt dθ ,dt 2 by requiring (P h, k u , v) = (u , v) for all v ∈ Sh,k (RT ). We have also the dθ ,dt dθ dθ dt dt representation P h, k = P h ⊗ P k = P k ⊗ P h . The low order projections have the approximation property s, s 1,0 s−r r (1.50) ku − P h, + k (s−r)/2 kuks, 2s , (u ∈ H002 (R2T )), k ukr, 2 ≤ c h

where 0 ≤ r ≤ s ≤ 2, r < 1. For the higher order cases, we restrict ourselves to the following basic bounds 1,1 s−r r ku − P h, + k (s−r)/2 kuks, 2s , (0 ≤ r ≤ 2, s = 1 or s = 2), (1.51) k ukr, 2 ≤ c h 3,1 s−r (s−r)/2 ku − P h, k ukr, r2 ≤ c h +k kuks, 2s , (0 ≤ r ≤ 2, s = 2 or s = 4), (1.52) s, s

for u ∈ H002 (R2T ). Compare also [14], and [37]. The spline interpolation makes sense only when the associated functions are continuous. According to the Sobolev embedding theorem this is guaranteed if u ∈ H r , r > 12 . The interpolation operator Ihdθ : H r → Shdθ , r > 12 , is defined as

Ihdθ u ∈ Shdθ (Ihdθ u)(θ˜i ) = u(θ˜i ), i = 1, 2, . . . , N,

where θ˜i are the usual interpolation points θi for all i, when dθ is odd θ˜i = 1 (θ + θ ) for all i, when dθ is even. i−1 i 2 In this work, we apply only odd degree interpolation. In the periodic case, we have for quasi-uniform meshes the approximation property [21] ku − Ihdθ ukH r ≤ c hs−r kukH s ,

(0 ≤ r ≤ s ≤ 2, r < dθ + 12 , s > 12 ).

(1.53)

Our Petrov-Galerkin approach is to use interpolation with respect to the space variable and L2 -projection with respect to the time variable. For this, we define the θ ,dt two-dimensional projection operator Qdh,k := Ihdθ ⊗P kdt = P kdt ⊗Ihdθ . The following approximation property is used for the analysis of the nonlinear problems. Lemma 1 Let the meshes be quasi-uniform. Then we have the estimates ku − Q1,0 h,k uk0,0

≤

ku − Q1,0 h,k uk 12 , 14

≤

s s, s c hs + k 2 kuks, 2s , (u ∈ H002 (R2T )), 12 < s ≤ 2),(1.54) 1 1 1, 1 c h 2 + k 4 kuk1, 12 , (u ∈ H00 2 (R2T )). (1.55)

27 Proof.

We use the decomposition 0 0 1 (I − Q1,0 h,k )u = (I − P k )u − P k (I − Ih )u

(1.56)

and bound the H 0,0 -norm of the terms separately. For the first term, we have k(I −

P k0 )uk20,0

≤

Z1

c k 2r ku(θ, ·)k2H r (0,T ) dθ = c k 2r kuk20,r ,

(0 ≤ r ≤ 2). (1.57)

0

A similar estimate combined with the L2 -stability of the P k0 -operator gives kP k0 (I

−

Ih1 )uk20,0

≤c

ZT

k(I − Ih1 )u(·, t)k2H 0 dt = c h2r kuk2r,0 , ( 21 < r ≤ 2). (1.58)

0

Estimate (1.54) for (1.58)

1 2

< s ≤ 2 follows by putting together (1.56) with (1.57) and s

s

s s 2 2 s s ku − Q1,0 h,k uk0,0 ≤ c k kuk0, 2 + c h kuks,0 ≤ c ( h + k )kuks, 2 .

Another decomposition, namely 1,0 1,0 1,0 u − Q1,0 h,k u = (u − P h, k u) + (P h, k u − Qh,k u),

inverse estimate (1.45) and the approximation properties of the projection operators yield (1.55).

1.7 Some Petrov-Galerkin approximations For the single layer heat operator equation Su = f , we have analysed several methods. These include collocation methods, as well as Petrov-Galerkin type methods, where collocation is used with respect to the space variable, and the Galerkin method with respect to the time variable. In the case of the hypersingular heat operator equation Hu = f , only the latter approach is used. The purpose of this chapter is to describe a unified approach which covers the treatment given in articles I, II, and III in the light of Theorem 11. An essential device is the equivalence of the proposed projection method to a certain Galerkin method. In Rorder to establish this equivalence, we need the mean 1 value functional (J w)(t) := 0 w(θ, t) dθ, as well as the corresponding trapezoidal rule approximation (J∆ w)(t) :=

N X θn+1 − θn−1 w(θn , t). 2 n=1

(1.59)

28 We define the bilinear forms B jθ ,jt (w, v) jθ ,jt B∆ (w, v)

=

∂tjt (∂θjθ + J )w , ∂tjt (∂θjθ + J )v

=

∂tjt (∂θjθ

+ J∆ )w ,

∂tjt (∂θjθ

(1.60)

+ J )v .

(1.61)

In particular, we interpret here ∂t0 u = u. Throughout this work, the index jθ is related to the degree of the spline approximation by the formula dθ = 2jθ − 1. With respect to the time index, this convention is not applicable. In practice, jt takes the values jt = 0 or jt = 1. Using the orthogonality conditions (∂θs u , J u) = 0, (∂θs u , J∆ u) = 0, s = 1, 2, . . . , we may rewrite the bilinear forms B jθ ,jt (w, v)

=

jθ ,jt B∆ (w, v)

=

∂tjt ∂θjθ w , ∂tjt ∂θjθ v + ∂tjt J w , ∂tjt J v ∂tjt ∂θjθ w , ∂tjt ∂θjθ v + ∂tjt J∆ w , ∂tjt J v .

The next step is to define a family of seminorms. For brevity, we introduce the notation |||v|||jθ ,jt ;r = k∂tjt ∂θjθ vkr, r2 + k∂tjt J vkr, r2 . (1.62) The indices jθ and jt depend on the degree of the approximating tensor product splines. The third index r depends on the order of the operator under consideration. In the cases of the single layer heat operator equation and the hypersingular heat operator equation, the index r takes the values r = − 12 , r = + 12 , respectively. In any space of functions, where the expression ||| · |||jθ ,jt ;r is well defined and finite, it gives a seminorm. However, in the subspace of the approximating functions it defines a norm. This is proved in the following Lemma. Lemma 2 Let r ∈ R, dθ = 2jθ − 1, and jt = 0 or jt = 1. Then the mapping dθ ,dt v 7→ |||v|||jθ ,jt ;r defines a norm in the space Sh,k (R2T ). Proof.

It suffices to show that the condition |||v|||jθ ,jt ;r = 0,

dθ ,dt for v ∈ Sh,k (R2T )

(1.63)

implies v(θ, t) ≡ 0. We consider only the case jt = 1 (dt = 1), since the other case jt = 0 (dt = 0) is a simplified version of the proof. Assuming (1.63), we obtain from the basis representation (∂t ∂θjθ v)(θ, t)

= ∂t

X M X N

αm,n (∂θjθ ψndθ )(θ)φdmt (t)

X M X N

αm,n (J ψndθ )φdmt (t) .

(1.64)

m=1 n=1

and (∂t J v)(t) = ∂t

m=1 n=1

(1.65)

29 According to (1.63) and (1.64) we have which, because of the initial condition N X

M P

N P

m=1 n=1 φ1m (t) = 0

αm,n (∂θjθ ψndθ )(θ) ≡ 0,

αm,n (∂θjθ ψndθ )(θ)

φdmt (t) ≡ C(θ),

for all m = 1, . . . , M , yields

(m = 1, . . . , M ).

(1.66)

n=1

Analogously, from (1.63) and (1.65), we deduce that N X

αm,n (J ψndθ ) = 0,

(m = 1, . . . , M ).

(1.67)

n=1

Adding (1.66) and (1.67) together gives N X

αm,n (∂θjθ + J )ψndθ = (∂θjθ + J )(

n=1

N X

αm,n ψndθ ) ≡ 0,

(m = 1, . . . , M ).

n=1 (dθ −1)/2

The operator (∂θjθ + J ) is an isomorphism from Shdθ to the space Sh N X

αm,n ψndθ = 0,

. Thus

(m = 1, . . . , M ),

n=1

which implies that all the coefficients αm,n vanish, proving our statement. After these preparations, we are able to reformulate the proposed projection methods as certain Galerkin methods. This equivalence is carried by the approximating bilinear form defined in (1.61). Our next aim is to establish the coercivity of the approximating bilinear form with respect to the associated energy norm ||| · |||jθ ,jt ;r . For our method, it is enough that this property is valid in the subspace dθ ,dt Sh,k (R2T ).

1.7.1 Reformulation of the collocation problem The lowest order spline collocation equations corresponding to the single layer heat operator equation are the following: dθ ,1 dθ ,1 dθ ,1 find u∆ ∈ Sh,k such that Ih,k Su∆ = Ih,k f,

dθ = 1 or dθ = 3.

(1.68)

For a sufficiently smooth solution to the equation Su = f , the collocation problem is equivalent to the Galerkin problem: dθ ,1 jθ ,1 jθ ,1 find u∆ ∈ Sh,k such that B∆ (Su∆ , v) = B∆ (Su, v).

(1.69)

The case dθ = 1 was analysed by Hamina & Saranen in paper I for a circular boundary. Their regularity requirements were described in terms of the spaces of

30 continuous functions. Later Hämäläinen & Saranen [32] and H¨ am¨ al¨ ainen [31] reduced the regularity requirements by introducing appropriate spaces of the Sobolev type. They also generalized the analysis to cover the case of a general smooth boundary curve, provided that the parametric representation is with respect to the arc length parameter. The case dθ = 3 has not been explicitly treated in the articles.

1.7.2 Reformulation of the Petrov-Galerkin method θ ,dt Another possibility is to use the projection operator Qdh,k . Then the lowest order approximations to the single layer heat operator equation are the following.

1,0 Find u∆ ∈ Sh,k (R2T )

Find u∆ ∈ Find u∆ ∈

1,1 Sh,k (R2T ) 3,1 Sh,k (R2T )

such that

1,0 Q1,0 h,k Su∆ = Qh,k f.

(1.70)

such that

Q1,1 h,k Su∆ Q3,1 h,k Su∆

(1.71)

such that

= =

Q1,1 h,k f. Q3,1 h,k f.

(1.72)

The equivalent Galerkin method is now available. s, s

Theorem 9 Let u ∈ H002 (R2T ), s > 12 be the solution of the equation Su = f . 1,0 The function u∆ ∈ Sh,k (R2T ) is a solution of problem (1.70) if and only if 1,0 1,0 B∆ (Su∆ , v) = B∆ (Su, v),

1,0 (v ∈ Sh,k (R2T )).

(1.73)

1,1 The function u∆ ∈ Sh,k (R2T ) is a solution of problem (1.71) if and only if 1,0 1,0 B∆ (Su∆ , v) = B∆ (Su, v),

1,1 (v ∈ Sh,k (R2T )).

(1.74)

s, s

Let u ∈ H002 (R2T ), s > 32 be the solution of the equation Su = f . Then the 3,1 function u∆ ∈ Sh,k (R2T ) is a solution of problem (1.72) if and only if 2,0 2,0 B∆ (Su∆ , v) = B∆ (Su, v),

3,1 (v ∈ Sh,k (R2T )).

(1.75)

The proof is similar to that of Theorem 10, and is omitted. It is remarkable here that the required regularity for the equivalence is described in terms of the anisotropic Sobolev spaces contrary to article II, where the spaces of essentially bounded functions were used. See also Theorem 18, formula (3.4). The hypersingular heat operator decreases smoothness. Therefore, we have considered only the lowest order method corresponding to the projection operator θ ,dt Qdh,k . The discrete approximation to the hypersingular heat operator equation is: 3,1 3,1 find u∆ ∈ Sh,k (R2T ) such that Q3,1 (1.76) h,k Hu∆ = Qh,k f. This operator equation is equivalent to the following system of equations:

31 3,1 find u∆ ∈ Sh,k (R2T ) such that

ZT

ZT φ1m (t)(Hu∆ )(θn , t) dt = φ1m (t)f (θn , t) dt, (n = 1, . . . , N, m = 1, . . . , M ).

0

0

(1.77) As an example, we prove that problem setting (1.76) is equivalent to a certain Galerkin problem. In this particular case, the bilinear forms are B 2,0 (w, v) = ∂θ2 w , ∂θ2 v + J w , J v (1.78) 2,0 2 2 B∆ (w, v) = ∂θ w , ∂θ v + J∆ w , J v . (1.79) The proof is quoted from paper III. s, s

Theorem 10 Let u ∈ H002 (R2T ), s > 52 be the solution of the equation Hu = f . 3,1 Then the function u∆ ∈ Sh,k (R2T ) is a solution of problem (1.76) if and only if 2,0 2,0 B∆ (Hu∆ , v) = B∆ (Hu, v),

Proof.

3,1 (v ∈ Sh,k (R2T )).

(1.80)

We denote w ˜ = Hu∆ − f, and consider the condition 2,0 B∆ (w, ˜ v) = 0,

3,1 for all v ∈ Sh,k (R2T ).

(1.81)

It is known that the operator ∂θ2 + J : S 3h → S 1h is an isomorphism between the one-dimensional periodic spline spaces. Therefore, we can choose the basis 3,1 functions vm,n of the trial space Sh,k (R2T ) such that (∂θ2 + J )vm,n (θ, t) = φ1m (t)ψn1 (θ), where ψn1 is the 1-periodic Courant basis function,   1 + (θ − θn )/(θn − θn−1 ), 1 − (θ − θn )/(θn+1 − θn ), ψn1 (θ) =  0,

such that for θ ∈ [0, 1] θn−1 < θ < θn , θn < θ < θn+1 , otherwise.

Then we obtain by partial integration 2,0 B∆ (w, ˜ vm,n ) = (∂θ2 + J∆ )w ˜ , (∂θ2 + J )vm,n = (∂θ2 + J∆ )w ˜ , φ1m ψn1 = − ∂θ w ˜ , φ1m ∂θ ψn1 + J∆ w ˜ , φ1m ψn1 =−

RT 0

w(θ ˜ n+1 , t)− w(θ ˜ n , t) w(θ ˜ n , t)− w(θ ˜ n−1 , t) − dt θn − θn−1 θn+1 − θn RT 1 PN n−1 + J ψn1 n=1 θn+1 −θ φm (t)w(θ ˜ n , t) dt. 2

φ1m (t)

0

(1.82)

32 Suppose first that u∆ satisfies the discrete equations, which means that ZT

φ1m (t)w(θ ˜ n , t) dt = 0,

for all n = 1, . . . , N, m = 1, . . . , M.

(1.83)

0 2,0 According to (1.82), the bilinear form B∆ (w, ˜ v) reduces to a linear combination 2,0 3,1 of integrals of type (1.83) and consequently B∆ (w, ˜ v) = 0 for all v ∈ Sh,k (R2T ). 3,1 3,1 Conversely, assume that (1.81) is valid. For all v ∈ Sh,k (R2T ) also J v ∈ Sh,k (R2T ). RT 2,0 The identity B∆ (w, ˜ Jv) = 0 (J∆ w)(t)(J ˜ v)(t) dt together with (1.79) implies

ZT

(J∆ w)(t)(J ˜ v)(t) dt

= 0,

3,1 (v ∈ Sh,k (R2T )),

(1.84)

(∂θ2 w ˜ , ∂θ2 v)

= 0,

3,1 (v ∈ Sh,k (R2T )).

(1.85)

0

As in (1.82), formula (1.85) yields Z

T

φ1m (t)

0

w(θ ˜ n+1 , t) − w(θ ˜ n , t) w(θ ˜ n , t) − w(θ ˜ n−1 , t) − dt = 0 θn+1 − θn θn − θn−1

(1.86)

for 1 ≤ n ≤ N, 1 ≤ m ≤ M . Let m be fixed. Then we have for all n = 1, ..., N Z T − w(θ ˜ n , t) w(θ ˜ n , t) − w(θ ˜ n−1 , t) dt = φ1m (t) dt = Cm . − θn θ − θ n n−1 0 (1.87) Using the periodicity of w ˜ with respect to θ, we obtain Z

T

w(θ ˜ n+1 , t) φ1m (t) θn+1 0

N X

(w(θ ˜ n , t) − w(θ ˜ n−1 , t)) = w(θ ˜ N , t) − w(θ ˜ 0 , t) = 0.

(1.88)

n=1

On the other hand, (1.87) implies

Cm

=

Cm

N X

T

(θn − θn−1 ) =

n=1

=

ZT 0

φ1m (t)

N Z X

φ1m (t)(w(θ ˜ n , t) − w(θ ˜ n−1 , t)) dt

n=1 0 N X

(w(θ ˜ n , t) − w(θ ˜ n−1 , t)) dt,

n=1

which, with (1.88), gives Cm = 0. Inserting this back to (1.87), we have ZT 0

φ1m (t)w(θ ˜ n , t) dt

=

ZT 0

φ1m (t)w(θ ˜ 0 , t) dt,

(1 ≤ n ≤ N ).

(1.89)

33 Finally, according to (1.84) and (1.89)

0=J

ψn3

ZT ZT N X θn+1 − θn−1 1 3 φm (t)w(θ ˜ n , t) dt = J ψn φ1m (t)w(θ ˜ 0 , t) dt. (1.90) 2 n=1 0

0

Since J ψn3 is nonzero, the value of the integral

RT

φ1m (t)w(θ ˜ 0 , t) dt vanishes and, due

0

to formula (1.89), all integrals of type (1.83) vanish. Thus, u∆ satisfies PetrovGalerkin equations (1.76).

1.8 Description of the nonlinearity Now, we specify the assumptions on the nonlinearity. The real valued nonlinear function F (x, t, ξ) : ΣT ×R → R is assumed to satisfy the Carathéodory conditions: The mapping (x, t) 7→ F (x, t, ξ) : ΣT → R is measurable for all fixed ξ ∈ R.(1.91) The mapping ξ 7→ F (x, t, ξ) : R → R is continuous for almost all (x, t) ∈ ΣT(1.92) . The associated Nemitsky operator or superposition operator u 7→ F(u) : L2 (ΣT ) → L2 (ΣT );

F(u)(x, t) = F (x, t, u(x, t))

is well defined, provided that the Carathéodory conditions and the following growth condition |F (x, t, ξ)| ≤ b(x, t) + c |ξ|, ((x, t, ξ) ∈ ΣT × R) (1.93) are valid (see [38]). Here, c is a constant and b(x, t) ∈ L2 (ΣT ). For the analysis of the numerical approximation scheme, we assume that the mapping ξ 7→ F (x, t, ξ) is nondecreasing for each (x, t) ∈ ΣT (F (x, t, ξ) − F (x, t, ξ 0 ))(ξ − ξ 0 ) ≥ 0

for all

ξ, ξ 0 ∈ R.

(1.94)

Note that the Carathéodory conditions, together with (1.93) and (1.94) imply that the Nemitsky operator F : L2 (ΣT ) → L2 (ΣT ) is monotone, which by definition means that (F(u)−F(w) , u − w) ≥ 0.

34 In order to obtain regularity results and error estimates, we need the Lipschitz and H¨ older conditions. There exists a constant L > 0 such that |F (x, t, ξ) − F (x, t, ξ 0 )| ≤ L|ξ − ξ 0 |

for all ξ, ξ 0 ∈ R, (x, t) ∈ ΣT .

(1.95)

There are constants N, M > 0 and 0 < λ ≤ 1 such that λ

|F (x, t, ξ) − F (y, τ, ξ)| ≤ N |x − y|λ + M |t − τ | 2

(1.96)

for all ξ ∈ R, (x, t), (y, τ ) ∈ ΣT . We recall some basic properties concerning the Nemitsky operator in anisotropic Sobolev spaces. The first one is a special case of the results given for example in book [38]. The latter is presented in [30]. For the convenience of the reader, the proofs are repeated here. Lemma 3 Assume that the Carathéodory conditions are valid. (i) If (1.93) is valid, then F : L2 (ΣT ) → L2 (ΣT ) is bounded and continuous. (ii) If, in addition (1.94) is valid, then the Nemitsky operator F : L2 (ΣT ) → L2 (ΣT ) is monotone. Proof.

Assuming growth condition (1.93), we obtain

kF(u)k2L2 (ΣT )

=

ZT Z

≤

ZT Z

0

0

2

|(F(u))(x, t)| dΓx dt =

ZT Z 0

Γ

|F (x, t, u(x, t))|2 dΓx dt

Γ

(b(x, t) + c|u(x, t)|)2 dΓx dt

Γ

≤ kbk2L2 (ΣT ) + c kuk2L2 (ΣT ) . The monotonicity of the Nemitsky operator follows because

(F(u)−F(w) , u−w) =

ZT Z 0

is non-negative.

(F (x, t, u(x, t))−F (x, t, w(x, t)))(u(x, t)−w(x, t))dΓx dt

Γ

Lemma 4 Assume that Carathéodory conditions are valid. (i) If (1.93) and (1.95) are valid, then F : L2 (ΣT ) → L2 (ΣT ) is Lipschitz continuous. s, s

s, s

(ii) If (1.93),(1.95) and (1.96) are valid, then F : H002 (ΣT ) → H002 (ΣT ) is bounded for all 0 ≤ s < λ.

35 Proof.

r, r2

The norm of the space H00 (ΣT ) is defined by kuk2s, 2s = kuk2L2 (ΣT ) + |u|2s,0 + |u|20, 2s ,

where the seminorms are defined by |u|2s,0

=

ZT Z Z

=

ZT ZT Z

0

|u|20, 2s

Γ Γ

0

0

Γ

|u(x, t) − u(y, t)|2 dΓx dΓy dt, |x − y|1+2s |u(x, t) − u(x, τ )|2 dΓx dt dτ . |t − τ |1+s

(i) The Lipschitz continuity of the Nemitsky operator follows from condition (1.95) kF(u) − F(w)k2L2 (ΣT )

=

RT R

|(F(u))(x, t) − (F(w))(x, t)|2 dΓx dt

=

RT R

|F (x, t, u(x, t)) − F (x, t, w(x, t))|2 dΓx dt

0 Γ

≤

0 Γ RT R

L2 |u(x, t) − w(x, t)|2 dΓx dt

0 Γ 2

= L ku − wk2L2 (ΣT ) . (ii) The Lipschitz and Hölder conditions for the function F = F (x, t, ξ) yield for the seminorms the estimates RT R R |(F (u))(x,t)−(F (u))(y,t)|2 |F(u)|2s,0 = dΓx dΓy dt |x−y|1+2s 0 Γ Γ

≤

RT R R |F (x,t,u(x,t))−F (y,t,u(x,t))|2

≤

RT R R

|x−y|1+2s

0ΓΓ

N 2 |x−y|2λ |x−y|1+2s

0 Γ ΓR

≤ N 2T

R

(y,t,u(y,t))|2 + |F (y,t,u(x,t))−F dΓx dΓy dt |x−y|1+2s

dΓx dΓy dt + L2 |u|2s,0

|x − y|2λ−2s−1 dΓx dΓy + L2 |u|2s,0 ≤ c1 + L2 |u|2s,0

Γ Γ

|F(u)|20, s = 2

≤ =

RT RT R

0 0 Γ RT RT R 0 0 Γ RT RT R 0 0 Γ

|(F (u))(x,t)−(F (u))(x,τ )|2 dΓx |t−τ |1+s

dt dτ

|F (x,t,u(x,t))−F (x,τ,u(x,t))|2

+

|t−τ |1+s

M 2 |t−τ |λ |t−τ |1+s

= M 2 m(Γ)

RT RT

L2 |u(x,t)−u(x,τ )|2 dΓx |t−τ |1+s

dt dτ

dΓx dt dτ + L2 |u|20, s

2

|t − τ |λ−s−1 dt dτ + L2 |u|20, s = c2 + L2 |u|20, s . 2

0 0

2

The assertion follows since kF(u)k2s, s = kF(u)k2L2 (ΣT ) + |F(u)|2s,0 + |F(u)|20, s 2 2 ≤ kbk2L2 (ΣT ) + c1 + c2 + c kuk2L2 (ΣT ) + L2 |u|2s,0 + L2 |u|20, s 2 ≤ c3 + c4 kuk2s, s . 2

2 Abstract theory The theory of the Finite Element Method, as well as that of the Boundary Element Method, is closely related to the abstract functional analysis. The so-called LaxMilgram theorem, together with the underlying Riesz representation theorem, play crucial roles in this context. In this chapter, we introduce some fundamental elements of the linear and nonlinear theory. In the nonlinear case, the essential references are Petryshyn’s articles [41], [42] and [43], Minty’s work [36], and Browder’s results [11], [12]. The reader may find the textbooks by Pascali & Sburlan [38] and Deimling [19] useful.

2.1 Linear equations Let X be a Hilbert space. Let ` : X → R be a bounded linear functional. Let B(· , ·) : X × X → R be a bilinear form. The bilinear form is called bounded, if |B(w, v)| ≤ CB kwkX kvkX , for all w, v ∈ X, strongly coercive, if B(w, w) ≥ c0 kwk2X , for all w ∈ X, with c0 > 0. We consider the variational problem of finding u ∈ X, such that for all v ∈ X.

B(u, v) = `(v)

(2.1)

The solvability is given in the following so-called Lax-Milgram Theorem. For the proof, see [10] or [13]. Theorem 11 Let B(·, ·) be a real valued, strongly coercive and bounded bilinear form in a Hilbert space X. Let ` be a bounded real valued linear form on X. Then the equation B(u, v) = `(v)

for all v ∈ X

is uniquely solvable, and kukX ≤

1 k`k. c0

37 In order to approximate the solution of the variational problem (2.1), we consider another variational problem of finding un ∈ Xn such that for all vn ∈ Xn ,

B(un , vn ) = `(vn )

(2.2)

where Xn ⊂ X is a closed subspace. We recall the so-called Cea Lemma, which describes the solvability of problem (2.2). Theorem 12 Let B(·, ·) be a real valued and bounded bilinear form in a Hilbert space X. Let Xn be a closed subspace of X. Let B(·, ·) be strongly coercive in Xn . Let ` be a bounded real valued linear form on X. Then equation (2.2) admits a unique solution such that cB kukX . kun kX ≤ c0 For the approximation error in un , we have the quasi-optimal bound ku − un kX ≤ 1 +

cB inf ku − ψkX . c0 ψ∈Xn

Proof. The solvability of the variational problem (2.2) follows from the LaxMilgram theorem, since closed subspaces of Hilbert spaces are themselves Hilbert spaces. The coercivity and boundedness of the bilinear form combined to the equations B(u, vn ) = `(vn ) B(un , vn ) = `(vn )

for all vn ∈ Xn for all vn ∈ Xn

give c0 kun k2X0

≤

B(un , un ) = B(u, un ) ≤ cB kukX0 kun kX .

This yields the stability kun kX ≤

cB kukX . c0

In particular, we have for all ψ ∈ Xn c0 kun − ψk2X

≤ B(un − ψ, un − ψ) = B(u − ψ, un − ψ) + B(un − u, un − ψ) = B(u − ψ, un − ψ) ≤ cB ku − ψkX kun − ψkX

implying kun − ψkX ≤

cB inf ku − ψkX . c0 ψ∈Xn

Now, the triangle inequality gives for all ψ ∈ Xn ku − un kX ≤ ku − ψkX + kψ − un kX ≤ 1 + proving the quasi-optimality.

cB inf ku − ψkX c0 ψ∈Xn

38 In practice, spaces Xn are finite dimensional. The elements un ∈ Xn are called shape functions. The importance of the quasi-optimal error estimate is that the asymptotic convergence depends only on the approximation properties of the shape functions. In this study we apply these fundamental tools to the case of small perturbations of coercive operators. The formulation of Theorems 11 and 12 can be generalized in several ways. It is possible to replace the coercivity by the Babuska stability condition [10]. The theorems can also be modified to cover the case of compact perturbations of isomorphic operators [13], Theorem 10.1.3.

2.2 Nonlinear equations The purpose of this section is to give the general theoretical framework which has been applied to articles IV and V. The approach is based on Saranen’s article [55] concerning projection methods applied to the equation A(u) = g, where A : X → X is a nonlinear Hammerstein operator A(u) = Iu + Du + S(F (u)). Here, S and D are given linear operators and F describes the nonlinearity. Let X0 be a Hilbert space equipped with the norm k · kX0 such that the embedding X ⊂ X0 is dense and continuous. We denote the dual space of X by X ∗ . By identifying X0 with its dual, we have X ⊂ X0 ⊂ X ∗ . Let (· , ·) be the continuous bilinear form defined in X × X ∗ or X ∗ × X such that it extends the inner product of the pivot space X0 . We use another Hilbert space Z such that Z ⊂ X with continuous injection. We recall some concepts concerning nonlinear behaviour. The nonlinearity F : X0 → X0 is called monotone if (F (u) − F (w) , u − w) ≥ 0, strongly monotone if (F (u) − F (w) , u − w) ≥ cF ku − wk2X0 . Let (S −1 )∗ : X → X ∗ be the adjoint of the operator S −1 : X ∗ → X defined by the relation (S −1 u∗ , u) = (u∗ , (S −1 )∗ u), u ∈ X. We assume that the operator A is strongly (S −1 )∗ -monotone in the sense that the inequality (A(u) − A(w) , (S −1 )∗ (u − w)) ≥ cA ku − wk2X

(2.3)

is valid for all u, v ∈ X with a positive constant cA . The monotonicity (2.3) implies that the operator A : X → X is a homeomorphism. Let Xn ⊂ X, n ∈ N be a sequence of finite dimensional subspaces with given projection Qn : X → Xn for each n ∈ N. We look for an approximation un for the solution u such that Qn A(un ) = Qn A(u). (2.4)

39 The following abstract result describes the solvability of the approximate equation. Theorem 13 We assume that the following conditions are fulfilled D: S:

X → Z is bounded, X ∗ → X is an isomorphism and S : X0 → Z is bounded,

(2.5) (2.6)

F : A:

X0 → X0 is Lipschitz continuous, X → X is strongly (S −1 )∗ − monotone.

(2.7) (2.8)

Assume that the projection operators Qn have the asymptotic approximation property ku − Qn ukX ≤ ε(n)kukZ , n ∈ N (2.9) such that ε(n) → 0 when n → ∞. Then there exists an integer n0 ∈ N and a positive constant c1 such that kQn A(v) − Qn A(v 0 )kX ≥ c1 kv − v 0 kX for all v, v 0 ∈ Xn ,

(2.10)

when n ≥ n0 . Moreover, equation (2.4) is uniquely solvable for n ≥ n0 . Proof.

Since Qn is a projection Qn v = v for all v ∈ Xn , and we can write

(I − Qn )(A(v) − A(v 0 )) = (I − Qn )D(v − v 0 ) + (I − Qn )S(F (v) − F (v 0 )), for all v, v 0 ∈ Xn . A lower bound is obtained by decomposing and using (S −1 )∗ -monotonicity Qn A(v) − Qn A(v 0 ), (S −1 )∗ (v − v 0) 0 −1 ∗ = A(v) − A(v 0 ), (S −1 )∗ (v − v 0 ) − (I − Q ) (v − v 0 ) n ) A(v)−A(v ) , (S ≥ c kv − v 0 k2X − c0 k(I − Qn ) A(v)− A(v 0 ) kX kv − v 0 kX . On the other hand, the Schwarz inequality yields the upper bound Qn A(v)−Qn A(v 0 ), (S −1 )∗ (v−v 0 ) ≤ c kQn A(v)−Qn A(v 0 )kX k(S −1 )∗ (v−v 0 )kX ∗ ≤ c kQn A(v)−Qn A(v 0 )kX kv − v 0 kX . Therefore, we obtain the estimate kQn A(v) − Qn A(v 0 )kX ≥ c kv − v 0 kX − c0 k(I − Qn ) A(v) − A(v 0 ) kX .

(2.11)

According to approximation property (2.9) combined with assumptions (2.5) (2.8), we find k(I −Qn ) A(v)−A(v 0 ) kX ≤ k(I −Qn )D(v−v 0 )kX + k(I −Qn )S(F (v)−F (v 0 ))kX ≤ c ε(n)kD(v − v 0 )kZ + c ε(n)kS(F (v)−F (v 0 ))kZ ≤ c ε(n)kv − v 0 kX + c ε(n)kF (v)−F (v 0 )kX0 ≤ c ε(n)kv − v 0 kX + c ε(n)kv − v 0 kX0 ≤ c ε(n)kv − v 0 kX . Thus, (2.11) yields the stability kQn A(v) − Qn A(v 0 )kX ≥ c1 kv − v 0 kX , for all v, v 0 and n ≥ n0 .

40 Due to stability, the operator Qn A : Xn → Xn is an injection. According to Brouwer’s theorem on invariance of the domain, the range R(Qn A) is open. Because of the stability and continuity of Qn A in Xn , the range R(Qn A) is also closed in Xn . Therefore, R(Qn A) = Xn , and Qn A is a homeomorphism. Theorem 14 We assume that conditions (2.5) - (2.8) and approximation property (2.9) are valid. Then, we have the asymptotic error estimate ku − un k ≤ c ku − Qn ukX .

(2.12)

Furthermore, if the mapping D : X0 → Z is bounded, then ku − un k ≤ c ku − Qn ukX0 . Proof.

(2.13)

For all v ∈ Xn , because of equation (2.4) it holds that

Qn A(un ) − Qn A(v) = Qn A(u) − Qn A(v)

(2.14)

= Qn (u − v) + Qn D(u − v) + Qn V (F (u)−F (v)). Since Qn is a projection operator in Xn , the choice v = Qn u gives Qn A(un ) − Qn A(Qn u) = Qn D(u − Qn u) + Qn V (F (u) − F (Qn u)).

(2.15)

Now, it follows by using (2.5) and approximation property (2.9) kQn D(u − Qn u)kX

≤ ≤

kD(u − Qn u)kX + k(I − Qn )D(u − Qn u)kX (2.16) c ku − Qn ukX + c ε(n)kD(u − Qn u)kZ

≤ ≤

c ku − Qn ukX + c ε(n)ku − Qn u)kX c ku − Qn ukX .

An analogous estimate follows by using (2.6) and (2.7) kQn V (F (u) − F (Qn u))kX ≤ kV (F (u) − F (Qn u))kX

(2.17)

+ k(I − Qn )V (F (u) − F (Qn u))kX ≤ c kF (u) − F (Qn u)kX0 + c ε(n)kV (F (u) − F (Qn u))kZ ≤ c ku − Qn ukX0 + c ε(n)kF (u) − F (Qn u)kX0 ≤ c ku − Qn ukX0 . Combining stability (2.10) with (2.15), (2.16) and (2.17) we have c1 kun − Qn ukX

≤ kQn A(un ) − Qn A(Qn u)kX ≤ c ku − Qn ukX + c kQn u − un kX .

Finally, the triangle inequality yields the convergence ku − un kX

≤

ku − Qn ukX + kQn u − un kX ≤ c ku − Qn ukX .

41 If mapping property (2.13) is valid then we may replace (2.16) by the stronger estimate kQn D(u − Qn u)kX ≤ ku − Qn ukX0 , which together with (2.10), (2.15) and (2.17) implies kun − Qn ukX ≤ c ku − Qn ukX0 . The theorem is proved.

3 Summary of the original articles The thesis consists of two lines of research. The first deals with questions of the stability and convergence of the spline collocation method applied to the single layer heat operator equation. These results are generalized to some spline PetrovGalerkin methods applied to the single layer heat operator equation, as well as to the case of the hypersingular heat operator equation. The analysis is based on the assumption that the spatial domain is a disk. The rest of the thesis consists of the analysis of the boundary element approximation of some problems, where the boundary condition is nonlinear. The effect of the numerical integration is analysed in the stationary case. Also some numerical results are given. Later, the theory is extended to the nonlinear time-dependent case.

3.1 Article I: On the spline collocation method for the single layer heat operator equation As discussed in the introduction, the direct boundary element method yields the single layer heat operator equation for the unknown boundary density. We construct and analyse a spline collocation scheme for solution of the single layer heat operator equation, assuming that the spatial domain is two-dimensional and has a smooth boundary. We apply piecewise linear approximation both in space and time together with nodal point collocation. For the proof of the stability and the convergence we restrict ourselves here to the case of the circle, but the method can be applied to all smooth closed curves, as was shown in [31], [32]. The collocation equations corresponding to the single layer heat operator equa1,1 tion Su = f are: find u∆ ∈ Sh,k such that (Su∆ )(θn , tm ) = f (θn , tm ),

(1 ≤ n ≤ N, 1 ≤ m ≤ M ).

(3.1)

For the equivalent characterization of the collocation equations, we need some 2 regularity assumptions. Let C1 (RT ) be the space of continuous functions f (θ, t)

43 2

on the closure RT of R2T such that f is 1-periodic with respect to the variable θ. Moreover, we define the spaces 2

C00 (RT ) 2 c H00 (RT )

2

= {f ∈ C1 (RT ) | f (θ, 0) ≡ 0}, 2 2 = {f ∈ C00 (RT ) | ∂θ f, ∂t f, ∂t ∂θ f = ∂θ ∂t f ∈ C1 (RT )}.

Next, we give sufficient conditions on the function u to guarantee the property 2 c Su ∈ H00 (RT ). Recall that the measurable function u is essentially bounded on 2 RT if there exists C > 0 such that |u(θ, t)| ≤ C for almost all (θ, t) ∈ R2T . We introduce the space t C00 (R2T ) = {f | f (θ, ·) is continuous for almost all θ and f (θ, 0) = 0}. t t The space H00 (R2T ) consists of functions f ∈ C00 (R2T ) such that ∂θ f ∈ C00 (R2T ) and the partial derivatives ∂t f, ∂t ∂θ f = ∂θ ∂t f are measurable, essentially bounded 2 1,1 is a subspace of H00 (R2T ). functions on RT . We emphasize that Sh,k Now, we are ready to establish the equivalence between the collocation problem and a certain Galerkin problem. For this, we define the bilinear form 1,1 B∆ (w, v) = ∂t ∂θ w , ∂t ∂θ v + (∂t ⊗ J∆ )w , (∂t ⊗ J)v R1 RT RT = ∂t ∂θ w(θ, t)∂t ∂θ v(θ, t) dt dθ+ ((∂t ⊗ J∆ )w)(t)((∂t ⊗ J)v)(t) dt. 0 0

0

Theorem 15 (I, Theorem 3.1) Let u ∈ H00 (R2T ) be the solution of the equation 1,1 Su = f . Then the function u∆ ∈ Sh,k is a solution of the collocation problem (3.1) if and only if 1,1 1,1 1,1 B∆ (Su∆ , v) = B∆ (Su, v), (v ∈ Sh,k ). The unique solvability of collocation equations (3.1) is a consequence of the 1,1 coercivity and continuity of the bilinear form B∆ (Su, v). This yields stability and convergence of the method. To prove the coercivity, we consider the form 1,1 B∆ (w, v) as a small perturbation of the bilinear form B 1,1 (w, v) = ∂t ∂θ w , ∂t ∂θ v + (∂t ⊗ J)w , (∂t ⊗ J)v . The energy norm induced by the bilinear form is |||v|||− 1 ,− 1 = k∂t ∂θ vk− 12 ,− 14 + k(∂t ⊗ J)vk− 12 ,− 14 . 2

4

1,1 For the analysis, it is essential that ||| · |||− 1 ,− 1 is a norm in the subspace Sh,k . 2 4 1,1 The proof of continuity and coercivity of the bilinear form B (S·, ·) is based on certain commutation properties of the single layer heat operator. The bilinear form 1,1 B∆ (S·, ·) is a small perturbation of the form B 1,1 (S·, ·). Therefore, we obtain the continuity estimate 1,1 |B∆ (Su, v)| ≤ c1 |||u|||− 1 ,− 1 |||v|||− 1 ,− 1 , 2

4

2

4

(u, v ∈ H00 (R2T )),

44 as well as the coercivity 2

1,1 B∆ (Sv, v) ≥ c2 |||v|||− 1 ,− 1 , 2

(v ∈ H00 (R2T )),

4

for a sufficiently small mesh parameter h. Now, the general theory gives the existence of the unique solution of collocation equations (3.1), as well as the stability |||u∆ |||− 1 ,− 1 ≤ 2

4

c1 |||u|||− 1 ,− 1 2 4 c2

and the quasi-optimal approximation result c1 inf1,1 |||u − v|||− 1 ,− 1 . |||u − u∆ |||− 1 ,− 1 ≤ 1 + 2 4 2 4 c2 v∈Sh,k

(3.2)

The convergence analysis of the collocation approximation defined by (3.1) is based on the quasi-optimal error estimate. It is natural first to discuss the asymptotic accuracy of the approximation when the error is measured by means of the energy norm ||| · |||− 1 ,− 1 . For this, we need error estimates for the L2 -orthogonal 2

4

1,1 1,1 2 2 projection P h, k : L (RT ) → Sh,k as well as some additional regularity assumptions on the solution of the single layer heat operator equation.

Lemma 5 Assume that u ∈ H00 (R2T ) such that u(θ, ·), (∂θ u)(θ, ·) ∈ H 2 (0, T ) for almost all θ, and (∂t u)(·, t) ∈ H 2 for almost all t. Then we have the approximation results 3 1 1 1,1 |||u − P h, ≤ c h 2 k∂t uk2,0 + c k h 2 + k 4 kuk0,2 + k∂θ uk0,2 , k u|||− 12 ,− 14 1,1 |||u − P h, ≤ c hk∂t uk2,0 + c k kuk0,2 + k∂θ uk0,2 . k u|||0,0 Theorem 16 (I, Theorem 5.1) Assume that the solution of the equation Su = f satisfies u ∈ H00 (R2T ) such that u(θ, ·), (∂θ u)(θ, ·) ∈ H 2 (0, T ) for almost all θ, 1,1 and (∂t u)(·, t) ∈ H 2 for almost all t. Then the collocation approximation u∆ ∈ Sh,k defined by (3.1) furnishes the asymptotic error estimate 3 1 1 |||u − u∆ |||− 1 ,− 1 ≤ c h 2 k∂t uk2,0 + c k h 2 + k 4 kuk0,2 + k∂θ uk0,2 . 2

4

In the estimate the time step dominates the order of the convergence. This effect can be compensated for by letting the time-discretization be finer than the discretization in the space variable. We also consider the convergence by using the norm |||v|||0,0 = k∂t ∂θ vk0,0 + k(∂t ⊗ J)vk0,0 . For this we need the inverse estimate 1

1

k∂t ∂θ vk0,0 + k(∂t ⊗ J)vk0,0 ≤ c max( h− 2 , k − 4 )|||v|||− 1 ,− 1 , 2

4

1,1 (v ∈ Sh,k ).

These aspects are summarized in the following theorem. Theorem 17 (I, Theorems 5.2, 5.3 and 5.4) Let the assumptions of Theorem 16 be valid. Moreover, suppose that k ≤ c hν , where h ≤ h0 is sufficiently small. Then the collocation approximation u∆ satisfies 5 1 3 |||u − u∆ |||− 1 ,− 1 ≤ c h 2 k∂t uk2,0 + c hmin( 4 ν,ν+ 2 ) (kuk0,2 + k∂θ uk0,2 . 2

4

45 In particular, for ν ≥

6 5

3 |||u − u∆ |||− 1 ,− 1 ≤ c h 2 kuk0,2 + k∂θ uk0,2 + k∂t uk2,0 . 2

4

If in addition, c0 h ≤ h ≤ h and c0 h2 ≤ k ≤ c h2 , where h ≤ h0 is sufficiently small. Then, we have for ν ≥ 65 |||u − u∆ |||0,0 ≤ c h kuk0,2 + k∂θ uk0,2 + k∂t uk2,0 . The final results are the pointwise and L2 -convergence 1 max |(u − u∆ )(θ, t)| ≤ c t 2 h kuk0,2 + k∂θ uk0,2 + k∂t uk2,0 , (0 ≤ t ≤ T ),

θ∈[0,1]

ku − u∆ k0,R2T ≤ c h kuk0,2 + k∂θ uk0,2 + k∂t uk2,0 . We have carried out some numerical experiments which confirm the convergence indicating, even the quadratic rate of the convergence. The numerical implementation of the scheme, covering also the general case, has been explained in some detail in article I.

3.2 Article II: A collocation type projection method for the single layer heat operator equation In order to simplify convergence analysis, and reduce regularity requirements, we propose another projection method for the approximation. Our approach is to use interpolation with respect to the space variable and L2 -projection with respect to 1 0 0 1 the time variable. With this aim in view, we define Q1,0 h,k := Ih ⊗ P k = P k ⊗ Ih , 1 1 r 1 where Ih : H → Sh , r > 2 , is the interpolation operator. Discrete equations 1,0 corresponding to the equation Su = f are: find u∆ ∈ Sh,k (R2T ) such that 1,0 Q1,0 h,k Su∆ = Qh,k f.

(3.3)

In order to describe the required regularity, we define some spaces of functions. 2 2 Let C1 (RT ) be the space of continuous functions f (θ, t) on the closure RT of R2T such that f is 1-periodic with respect to the variable θ. The measurable function u 2 is essentially bounded on RT if there exists a constant C > 0 such that |u(θ, t)| ≤ C for almost all (θ, t) ∈ R2T . The space of essentially bounded functions on R2T is denoted by L∞ (R2T ). Moreover, we define the spaces 2

C00 (RT ) 2

=

c H00 (RT ) = H00 (R2T ) =

2

{f ∈ C1 (RT ) | f (θ, 0) ≡ 0}, 2

2

{f ∈ C00 (RT ) | ∂θ f ∈ C1 (RT )}, {f ∈ L∞ (R2T ) | ∂θ f ∈ L∞ (R2T )}.

46 r

1,0 1,0 We notice that the inclusions Sh,k (R2T ) ⊂ H r, 2 (R2T ), r < 1, Sh,k (R2T ) ⊂ L∞ (R2T ) are valid for the trial spaces. Problem setting (3.3) is also equivalent to a certain Galerkin problem. For this, we define the bilinear form 1,0 B∆ (w, v)

= (∂θ w , ∂θ v) + (J∆ w , J v) R1 RT RT = ∂θ w(θ, t)∂θ v(θ, t) dt dθ + (J∆ w)(t)(J v)(t) dt, 0 0

0

which is a small perturbation of the bilinear form B 1,0 (w, v) = (∂θ w , ∂θ v) + (J w , J v). Theorem 18 (II, Theorem 2) Let u ∈ H00 (R2T ) be the solution of the equation 1,0 Su = f . Then the function u∆ ∈ Sh,k (R2T ) is a solution of discrete problem (3.3) if and only if 1,0 1,0 1,0 B∆ (Su∆ , v) = B∆ (Su, v), (v ∈ Sh,k (R2T )). (3.4) Here, the energy norm |||v|||− 1 ,− 1 = k∂θ vk− 12 ,− 14 + kJ vk− 12 ,− 14 is a norm in the 2

4

1,0 subspace Sh,k (R2T ). The continuity and coercivity of the form B 1,0 (S·, ·) follows from certain commutation properties of the single layer heat operator. The bilinear 1,0 form B∆ (S ·, ·) is a small perturbation. Therefore, we have, for sufficiently small h, the continuity 1,0 |B∆ (Su, v)| ≤ c1 |||u|||− 1 ,− 1 |||v|||− 1 ,− 1 , 2

4

2

4

(u, v ∈ H00 (R2T )).

(3.5)

and the coercivity 2

1,0 B∆ (Sv, v) ≥ c2 |||v|||− 1 ,− 1 , 2

4

(v ∈ H00 (R2T )).

(3.6)

We assume that the solution of the equation Su = f satisfies u ∈ H00 (R2T ). Then for all 0 < h ≤ h0 there exists a unique solution u∆ of equations (3.3). Moreover, we have the stability |||u∆ |||− 1 ,− 1 ≤ 2

4

c1 |||u|||− 1 ,− 1 2 4 c2

(3.7)

and the quasi-optimal error estimate |||u − u∆ |||− 1 ,− 1 ≤ 1 + 2

4

c1 inf |||u − v|||− 1 ,− 1 . 1,0 2 4 c2 v∈Sh,k (R2T )

(3.8)

The quasi-optimal error estimate (3.8) yields the asymptotic accuracy of the approximation when the error is measured by means of the norm ||| · |||− 1 ,− 1 . 2 4 For the proof of our convergence results, we need additional regularity assumptions on the solution of the equation Su = f . We also need approximation properties with respect to the norm |||v|||0,0 = k∂θ vk0,0 + kJ vk0,0 . These are described in the following lemma.

47 s, s

Lemma 6 Assume that u ∈ H00 (R2T ) such that u ∈ H002 (R2T ), 1 ≤ s ≤ 2. Then we have the approximation results 1,0 |||u − P h, k u|||− 1 ,− 1

≤

1,0 |||u − P h, k u|||0,0

≤

2

4

1

1

c ( hs− 2 + k (s− 2 )/2 )kuks, 2s , c hs−1 + k (s−1)/2 kuks, 2s ,

(1 ≤ s ≤ 2), (1 ≤ s ≤ 2).

¿From the quasi-optimal error estimate and Lemma 6, we obtain s, s

Theorem 19 (II, Theorem 4) Let Su = f , u ∈ H00 (R2T ) and u ∈ H002 (R2T ), 1,0 1 ≤ s ≤ 2. Then the approximation u∆ ∈ Sh,k (R2T ) defined by (3.3) furnishes the asymptotic error estimate 1

1

|||u − u∆ |||− 1 ,− 1 ≤ c ( hs− 2 + k (s− 2 )/2 )kuks, 2s . 2

4

(3.9)

In Theorem 19, the time step dominates the order of the convergence. This effect can be compensated for by letting the time-discretization be finer than the discretization in the space variable. Theorem 20 (II, Theorem 5 and Theorem 6) Let Su = f , u ∈ H00 (R2T ) s, s and u ∈ H002 (R2T ), 1 ≤ s ≤ 2. Moreover, suppose that h ≤ h and k ≤ c h2 , where h ≤ h0 is sufficiently small. Then the collocation type approximation u∆ satisfies 3 |||u − u∆ |||− 1 ,− 1 ≤ c h 2 kuk2,1 . (3.10) 2

4

Further, let c0 h ≤ h ≤ h and c0 h2 ≤ k ≤ c h2 , where h ≤ h0 is sufficiently small. Then we have for the collocation type approximation u∆ |||u − u∆ |||0,0 ≤ c hkuks, 2s .

(3.11)

In particular, we obtain the L2 -convergence ku − u∆ k0,0 ≤ c hkuk2,1 . We have proved that the Petrov-Galerkin approximation is stable and convergent. Compared to the collocation method, the regularity requirements are significantly reduced.

3.3 Article III: An approximation method for the hypersingular heat operator equation Here, we consider the case of a pseudodifferential operator of positive order. The model problem is the hypersingular heat operator equation. Our Petrov-Galerkin approach uses interpolation with respect to the space variable and L2 -projection 3 1 1 3 with respect to the time variable. For this, we define Q3,1 h,k := Ih ⊗ P k = P k ⊗ Ih , 1 3 r 3 where Ih : H → Sh , r > 2 , is the interpolation operator. Since our method uses pointwise values, it is essential that the image Hu is continuous. According to r, r the Sobolev embedding theorem this condition is true if u ∈ H002 (R2T ), r > 52 .

48 The discrete equation corresponding to the hypersingular heat operator equation 3,1 Hu = f is: find u∆ ∈ Sh,k (R2T ) such that 3,1 Q3,1 h,k Hu∆ = Qh,k f.

(3.12)

This operator equation is equivalent to the following system of equations: 3,1 find u∆ ∈ Sh,k (R2T ) such that ZT

φ1m (t)(Hu∆ )(θn , t) dt

ZT = φ1m (t)f (θn , t) dt, (n = 1, . . . , N, m = 1, . . . , M ).

0

0

Again, problem setting (3.12) is equivalent to a certain Galerkin problem. For this, we define the bilinear forms B 2,0 (w, v) = ∂θ2 w , ∂θ2 v + J w , J v 2,0 B∆ (w, v) = ∂θ2 w , ∂θ2 v + J∆ w , J v . s, s

Theorem 21 (III, Theorem 2) Let u ∈ H002 (R2T ), s > 52 be the solution of 3,1 the equation Hu = f . Then the function u∆ ∈ Sh,k (R2T ) is a solution of problem (3.12) if and only if 2,0 2,0 B∆ (Hu∆ , v) = B∆ (Hu, v),

3,1 (v ∈ Sh,k (R2T )).

(3.13)

Discrete equations (3.12) are uniquely solvable if the spatial discretization parameter is small enough. This result is a consequence of the coercivity esti2,0 mate for the bilinear form B∆ (Hu, v), which yields stability and convergence of the method. For the sake of brevity, we introduce the notation |||v|||r,s = k∂θ2 vkr,s + kJ vkr,s . In any space of functions, where this expression is well defined and finite, it gives a seminorm. In particular, in the subspace of the approximating functions ||| · |||s, s defines a norm. Our next aim is to establish the required 2 coercivity of the approximating bilinear form with respect to the norm ||| · ||| 1 , 1 . 2 4

3,1 For our method, it is enough that this property is valid in the subspace Sh,k (R2T ). s, s

Lemma 7 Let u, w ∈ H002 (R2T ), s > 52 . Then we have the continuity |B 2,0 (Hu, w)| ≤ c1 |||u||| 1 , 1 |||w||| 1 , 1 , 2 4

2 4

s, s

(u, w ∈ H002 (R2T ))

(3.14)

and the coercivity 2

B 2,0 (Hw, w) ≥ c2 |||w||| 1 , 1 , 2 4

s, s

(w ∈ H002 (R2T )).

(3.15)

The perturbation due to the trapezoidal rule approximation is small in the sense that 2,0 |B 2,0 (Hu, w) − B∆ (Hu, w)| ≤ c h |||u|||0,0 + |||u|||0, 1 |||w|||0,0 , (3.16) 2

49 s, 2s

for all u, w ∈ H00 (R2T ). In particular, 1 2 2,0 (Hv, v)| ≤ c h 1 + k − 4 |||v||| 1 , 1 , |B 2,0 (Hv, v) − B∆ 2 4

3,1 (v ∈ Sh,k (R2T )).

(3.17)

Therefore, also the approximating bilinear form is coercive 2

2,0 B∆ (Hv, v) ≥ c3 |||v||| 1 , 1 , 2 4

3,1 (v ∈ Sh,k (R2T ))

(3.18)

for 0 < h ≤ h0 , h4−ε ≤ c0 k, ε > 0 and h0 is sufficiently small. The main result is the following error bound. s, s

Theorem 22 (III, Theorem 3) Assume that u ∈ H002 (R2T ), s > 52 and Hu = f . Then, for all 0 < h ≤ h0 , h4−ε ≤ c0 k, ε > 0 and h0 is sufficiently small, there exists a unique solution u∆ of Petrov-Galerkin equations (3.12). Moreover, we have the approximation result |||u − u∆ ||| 1 , 1 ≤ C inf |||u − v||| (3.19) 1 1 + h|||u − v||| 1 . , 0, 3,1 2 4

2 4

v∈Sh,k (R2T )

2

The final aim is to establish L2 -convergence for the discrete solution. Based on the error estimate, we first discuss the asymptotic accuracy of the approximation when the error is measured by means of the norm ||| · ||| 1 , 1 . For the proof of the 2 4 convergence results, we make additional regularity assumptions on the solution of the equation Hu = f . 4,2 Lemma 8 Assume that u ∈ H00 (R2T ). Then we have the approximation results 3,1 |||u − P h, ≤ c h2 + k kuk4,2 , k u|||0,0 3,1 |||u − P h, k u||| 1 , 1 2 4

≤

3

3

c ( h 2 + k 4 )kuk4,2 .

¿From Theorem 22 and Lemma 8, we obtain Theorem 23 (III, Theorem 4 and Theorem 5) Let Hu = f , such that u ∈ 4,2 3,1 H00 (R2T ). Then the Petrov-Galerkin approximation u∆ ∈ Sh,k (R2T ) defined by (3.12) furnishes the asymptotic error estimate 3 3 |||u − u∆ ||| 1 , 1 ≤ c h 2 + k 4 kuk4,2 . (3.20) 2 4

4−ε

Moreover, suppose that h ≤ c0 k ≤ c h2 , where h ≤ h0 , and h0 is sufficiently small. Then we have the asymptotic estimates 3

|||u − u∆ ||| 1 , 1 ≤ c h 2 kuk4,2 ,

(3.21)

2 4

3

|||u − u∆ |||0,0 = k∂θ2 (u − u∆ )k0,0 + kJ (u − u∆ )k0,0 ≤ c h 2 kuk4,2 .

(3.22)

3 2

In particular, we have the L2 -convergence ku − u∆ k0,0 ≤ c h kuk4,2 . Note that the asymptotic order of convergence with respect to the |||·|||0,0 remains the same. It is remarkable here that the required regularity of the solution is described in terms of the anisotropic Sobolev spaces. This technique can be applied also to the case of the single layer heat operator equation.

50

3.4 Article IV: The numerical approximation of the solution of a nonlinear boundary integral equation with the collocation method In the stationary case, the nonlinear heat equation reduces to the potential equation (1.2) with a given nonlinear Neumann type boundary condition. The direct boundary integral approach yields the nonlinear boundary integral equation, [53] ( 12 I + DΓ )µ + SΓ F(µ) = SΓ gN Γ .

(3.23)

We denote the associated nonlinear integral operator A(u) = ( 12 I +DΓ )u+SΓ F(u). Here, F describes the nonlinearity, and the linear boundary integral operators are the single layer and double layer operators, respectively Z Z 1 1 ∂ u(y) ln |x − y|dΓy , DΓ u(x) = u(y) ln |x − y|dΓy . SΓ u(x) = − 2π 2π ∂ny Γ

Γ

The purpose of our paper is to introduce an approximation scheme for (3.23) by using an easily computable L2 -orthogonal projection of the nonlinear function. This approach applies to general projection methods, but for simplicity we discuss only collocation. Now, we specify the assumptions on the nonlinearity. The real valued nonlinear function F (x, ξ) : Γ × R → R is assumed to satisfy the Carathéodory conditions: The mapping x 7→ F (x, ξ) : Γ → R is measurable for all fixed ξ ∈ R. The mapping ξ 7→ F (x, ξ) : R → R is continuous for almost all x ∈ Γ. The Nemitsky operator u 7→ F(u) : L2 (Γ) → L2 (Γ); F(u)(x) = F (x, u(x)) is well defined, provided that the Carathéodory conditions and the growth condition |F (x, ξ)| ≤ b(x) + c |ξ|, (x, ξ) ∈ Γ × R are valid (see [38]). Here, c is a constant and b(x) ∈ L2 (Γ). For the analysis of the numerical approximation scheme, we assume that (A1) the Nemitsky operator F : L2 (Γ) → L2 (Γ) is strongly monotone, (A2) the Nemitsky operator F : L2 (Γ) → L2 (Γ) is Lipschitz continuous, (A3) the Nemitsky operator F : H s (Γ) → H s (Γ) is bounded for 0 ≤ s < 1. We point out that the Lipschitz continuity of the Carathéodory function F guarantees (A2) and (A3). These properties are necessary for regularity results and error estimates. Theorem 24 Let the capacity of the boundary curve differ from unity. 1

(i) For every g ∈ H − 2 (Γ) integral equation (3.23) has a unique solution u ∈ 1 H 2 (Γ). (ii) For the solution u, the following regularity result is true: If gN Γ ∈ H s−1 (Γ), 1 2 ≤ s < 2, and assumptions (A1), (A2) and (A3) are valid, then the solution satisfies u ∈ H s (Γ).

51 The proof is presented in [53]. It is based on the fact that integral operator A is strongly SΓ−1 monotone. Next, we consider the collocation method for finding an approximate solution of equation (3.23). We require that gN Γ ∈ H s−1 (Γ), s > 12 . Then the function gN Γ is continuous, and the collocation equations are given by: find uh ∈ Shd such that (A(uh ))(˜ xi ) = (SΓ gN Γ )(˜ xi ),

(i = 0, . . . , N − 1),

Nodal point collocation is applied for odd degree splines, and midpoint collocation is applied for even degree splines. An equivalent formulation of the collocation problem, given in terms of the interpolation operator is: find uh ∈ Shd such that Ihd A(uh ) = Ihd SΓ gN Γ ,

(3.24)

For numerical purposes, we define an approximate collocation equation as follows: find u ˜h ∈ Shd such that A˜h (˜ uh ) := 12 u ˜h − Ihd DΓ u ˜h + Ihd SΓ P hd F(˜ uh ) = Ihd SΓ gN Γ .

(3.25)

The following theorem describes the convergence properties. Theorem 25 (IV, Theorem 3.1 and Theorem 3.3) Assume d > 0. Let u ∈ H s (Γ), 21 < s ≤ d + 1, be the solution of (3.23) and suppose that (A1) and (A2) are valid. Then, for sufficiently small h, the collocation problem (3.24) as well as the approximate collocation problem (3.25) admit unique solutions uh , and u ˜h , respectively. Furthermore, we have the asymptotic error estimates ku − uh kt k˜ uh − uh k 12 ku − u ˜h kt

≤ ≤ ≤

c hs−t ||u||s , ch

s+ 12

(3.26)

kuks + c h

c hs−t kuks + c h

τ + 12

kF(u)kτ ,

τ +1−max(t, 12 )

kF(u)kτ ,

(3.27) (3.28)

for 0 ≤ t ≤ s, t < d + 12 , provided that F(u) ∈ H τ (Γ), 0 ≤ τ ≤ d + 1. The proof for (3.26) presented in [53] covers indices 12 ≤ t ≤ s, and the results in [55] give (3.26) when 0 ≤ t < 21 . Here, the crucial idea is to choose an appropriate Hilbert space, and show that the assumptions of Theorem 13 are fulfilled. Another source of error is the effect of the numerical integration. We decompose the single layer operator to a singular part with logarithmic singularity, and to a smooth part. The singular part is integrated exactly. The smooth part as well as integrals concerning the double layer operator are computed using numerical quadratures. We suppose that the quadrature error can be bounded above by c hσ . For the right hand side of the equation, we use the orthogonal projection approximation. The resulting nonlinear system defines the mapping Aˆ : Shd → Shd such that ˆ vh ) = Ih Ph gN Γ . A(ˆ (3.29) The solution vˆh is related to the approximate solution of the original problem by the formula vˆh = |x0 |ˆ uh . We summarize the final result in the following theorem. The details are written in the original article.

52 Theorem 26 (IV, Theorem 5.2) Assume d > 0. Let g ∈ H s−1 (Γ) and let v ∈ H s (Γ), 21 < s ≤ d + 1 be the solution of (3.29) and suppose that assumptions (A1), (A2) are valid. Then we have the estimate ku − u ˆ h kt

≤

1

1

c hs−t (kvks + kgks−1 ) + c hτ +1−max(t, 2 ) + c hσ−1−max(t, 2 ) ,

for 0 ≤ t ≤ s, t < d + 12 , provided that the data is sufficiently smooth. It evolves that our method retains the optimal convergence order of the collocation method. Numerical experiments confirm our theoretical results.

3.5 Article V: On the numerical solution of a non-linear heat conduction problem In this article, we analyse a projection method scheme for solution of the second kind non-linear heat operator equation (1.1), assuming that the spatial domain Ω is two-dimensional and has a smooth boundary Γ. The direct boundary integral approach yields the nonlinear boundary integral equation (1.21). We denote the associated nonlinear integral operator with A(u) = ( 12 I + DΓ )u + SΓ F(u)

(3.30)

and consider the nonlinear operator equation A(u) = SΓ gN Γ . Concerning the nonlinearity, we assume that Carathéodory conditions (1.91) and (1.92), growth condition (1.93) and Lipschitz condition (1.95) are true. These imply that the Nemitsky operator F : L2 (ΣT ) → L2 (ΣT ) is a well-defined, bounded and Lipschitz continuous operator. If, in addition, the mapping ξ 7→ F (x, t, ξ) is non-decreasing for each (x, t) ∈ ΣT then the Nemitsky operator is monotone. See Lemma 3 and Lemma 4. According to the following theorem, equation (3.30) is uniquely solvable 1 1

,

2 4 in H00 (ΣT ), [30]. 1 1

,

1 1

,

2 4 2 4 Theorem 27 The operator A : H00 (ΣT ) → H00 (ΣT ) is a homeomorphism. Furthermore, we have 1 1 2,4 A(u) − A(w) , SΓ−1 (u − w) Σ ≥ c ku − wk21 , 1 , (u, w ∈ H00 (ΣT )). T

2 4

We propose a method where we use interpolation with respect to the space variable, and orthogonal projection with respect to the time variable. The trial functions are tensor products of piecewise linear and piecewise constant splines. 1 0 0 1 We define the projection operator Q1,0 h,k := Ih ⊗P k = P k ⊗Ih . The discrete problem 1,0 asks us to find u∆ ∈ Sh,k such that 1,0 Q1,0 h,k A(u∆ ) = Qh,k SΓ gN Γ .

(3.31)

The method requires evaluation of pointwise values. Thus, we need a certain regularity concerning A(u∆ ) and SΓ gN Γ . For the right-hand side, it is enough to

53 s, 2s

assume that gN Γ ∈ H00 (ΣT ), s > 12 , since, according to the mapping property of s, s

the single layer operator, SΓ gN Γ ∈ H002 (ΣT ), s > 32 . Furthermore, it is known that r, r

the embedding H002 (ΣT ) ⊂ C(ΣT ) is continuous for r > 32 . Sufficient regularity for A(u∆ ) follows easily from the special structure of the operator A. The identity operator preserves splines and therefore the required pointwise values exist. On r, r 1,0 the other hand, the trial function space satisfies Sh,k ⊂ H002 (ΣT ), r < 1, and the mapping properties of the associated operators yield DΓ u∆ + SΓ F(u∆ ) ∈ C(ΣT ). 1,0 Lemma 9 If the meshes are quasi-uniform and v, v 0 ∈ Sh,k , then 0 k(I − Q1,0 h,k )DΓ (v − v )k 21 , 14

k(I −

Q1,0 h,k )SΓ (F(v)

0

− F(v ))k 12 , 14

≤ ≤

1 1 c h 2 + k 4 kv − v 0 k 12 , 14 , 1 c h + k 2 kv − v 0 k 12 , 14 .

(3.32) (3.33)

Proof. Estimate (3.33) follows from Lemma 1, with s = 1 by means of the mapping property of the single layer heat operator and the Lipschitz continuity of the Nemitsky operator 1 1 0 ≤ c h 2 + k 4 kSΓ (F(v) − F(v 0 ))k1, 12 k(I − Q1,0 h,k )SΓ (F(v) − F(v ))k 21 , 14 1 1 ≤ c h 2 + k 4 kF(v) − F(v 0 )k0,0 1 1 ≤ c h 2 + k 4 kv − v 0 k0,0 1 1 ≤ c h 2 + k 4 kv − v 0 k 12 , 14 . Estimate (3.33) is proved analogously.

Theorem 28 (V, Theorem 3) For sufficiently small h and k, we have the stability inequality 1,0 0 0 kQ1,0 h,k A(v) − Qh,k A(v )k 21 , 14 ≥ c0 kv − v k 12 , 14 ,

1,0 (v, v 0 ∈ Sh,k ).

In addition, equation (3.31) is uniquely solvable and we have the error estimates s ku − u∆ ks, 2s ≤ c inf1,0 ku − vks, 2s ≤ c h2−s + k 1− 2 kuk2,1 , (s = 0, s = 12 ). v∈Sh,k

Proof.

Since Q1,0 h,k is a projection, we can write 0 (I − Q1,0 h,k )(A(v) − A(v )) =

0 (I − Q1,0 h,k )DΓ (v − v )

+ (I −

1,0 Qh,k )SΓ (F(v)

(3.34) − F(v 0 )).

The Schwarz inequality 1,0 −1 0 0 Q1,0 h,k A(v)−Qh,k A(v ) , SΓ (v−v )

ΣT

1,0 0 0 ≤ c kQ1,0 h,k A(v)−Qh,k A(v )k 12 , 14 kv−v k 12 , 14

combined with the decomposition 1,0 −1 0 Q1,0 − v0 ) Σ h,k A(v) − Qh,k A(v ) , SΓ (v T −1 0 0 = A(v)− A(v 0 ) , SΓ−1 (v − v 0 ) Σ − (I − Q1,0 h,k ) A(v) − A(v ) , SΓ (v − v ) ΣT T 0 0 ≥ c kv − v 0 k21 , 1 − c0 k(I − Q1,0 h,k ) A(v)−A(v ) k 12 , 14 kv − v k 12 , 14 2 4

54 leads to the estimate 1,0 1,0 0 0 0 0 kQ1,0 h,k A(v)−Qh,k A(v )k 12 , 14 ≥ c kv−v k 12 , 14 − c k(I −Qh,k ) A(v)−A(v ) k 12 , 14 . This implies stability for 0 < h ≤ h0 and 0 < k ≤ k0 , since Lemma 9 together with (3.34) yields 1 1 0 0 2 4 k(I − Q1,0 h,k ) A(v) − A(v ) k 21 , 41 ≤ c ( h + k )kv − v k 12 , 14 . 1,0 1,0 Due to stability, the operator Q1,0 h,k A : Sh,k → Sh,k is an injection. According to Brouwer’s theorem on domain invariance, the range R(Q1,0 h,k A) is open. Because 1,0 1,0 of the stability and continuity of Qh,k A in Sh,k , the range R(Q1,0 h,k A) is also closed 1,0 1,0 1,0 in Sh,k . Therefore, R(Q1,0 A) = S , and Q A is a homeomorphism. The h,k h,k h,k quasi-optimality with respect to k · k 21 , 14 is a direct consequence of stability.

The L2 (ΣT )-estimate is proved by means of the technique in [55]. In fact, the result follows from Theorem 14, since conditions (2.5) - (2.8), and approximation 1 1

,

2 4 property (2.9) are true when we choose the spaces X0 = L2 (ΣT ), X = H00 (ΣT ),

1, 12

Z = H00 (ΣT ) and the operators S = SΓ , D = DΓ , F = F, Qn = Q1,0 h,k and 1,0 Xn = Sh,k . Also, the mapping property D : X0 → Z is bounded, is fulfilled. In the numerical implementation, the resulting time-integrals can be calculated exactly. Compared to the Galerkin method, this approach simplifies the realization of the matrix equations. It seems that straightforward application of the preceeding analysis does not extend to the collocation method.

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